In this blog post, I will study Question 3 from this test.

The question is about the maximum number of regions that a circle can be divided if we draw chords connecting points on its circumference.

Below I present the original question for your convenience:

Please do not read the rest of this article, if you want to attempt to solve this question on your own. The rest of this article describes my attempt at solving this question and you should not read it, unless you want to or you do not mind coming across relevant ideas, spoilers, hints, solutions, and strong opinions concerning this test.

You have been warned and I now consider that you continue to read knowing that what you come across for the rest of this article may forever spoil things for you and/or present strong opinions against this test.

Last warning: please do not read this blog post, unless you are certain that you know what you are doing. If you are not sure, then it would be best if you stopped reading at this point.

OK. If you are here, it means that you want to know my opinion. Well, ok then!

To cut a long story short, my opinion is that the test is highly inappropriate. In this blog post, I will focus on the study of question 3.

Well, any question from this test leaves me speechless, but this is not a compliment to the test. I find that the test is pointless, tedious, unintelligent and lacks explanation. I cannot get it. I cannot understand it. I do not know what to say. I am speechless.

Here we have this question. Question number 3. What does it want from us? Well, it is very clear what it wants. I wants us to find the maximum number of regions that are created when there are seven points on the circumference of a circle and we connect them with straight lines.

It is a nice problem and a known one, with a known solution. Since the authors of the test want it to maintain its integrity, why pose a well known problem? Points for originality: Nada! Since the authors of the test want to test intelligence, why ask a question that in this blog post I will prove that a toddler can answer? Points for intelligence: Nada! What is the purpose of such a question in a test for high intelligence? Points for sense and sensibility: Nada!

When I try to solve a question from the test, I feel to encounter two S’s. Solitude and Stupidity. Am I the Sole person who thinks that this test is Stupid?

Before we go any further, I would like to repeat that is a well known problem with a well known solution.

I do not where to begin, because this problem can be solved by a toddler, but more on this later.

First of all, this problem is discussed in the Martin Gardner’s book “Mathematical Circus”. Below I present the pages of this book that pertain to this problem. Also, at the end of the book there are references pertaining to this problem.

There are quite e few question from this exam that are depicted almost word-for-word in Martin Gardner’s books. As I said: Points for originality: Nada! I will let the reader decide who is the unoriginal: Martin Gardner or the authors of the test. At least, Martin Gardner names his sources!

So, here is the material from Martin Gardner’s book.

Below, I present two links that provide the solution (and the rationale behind it) to this well-known and well-studied problem.

But this problem can be solved by a toddler as well. The toddler may not be able to provide a general formula, but she will be able to count the number of regions. To prove to myself that a toddler could do it, I put myself in a toddler’s position. Although I did not crawl on all fours, (do not be surprised, that would be ridiculous for anyone else, but not for me!), I drew the relevant shape and counted the numbers of regions. Below, I present all cases from 2 points on the circumference up to 7 points on the circumference. Obviously, since we want the maximum number of regions, care has to be taken that each crossing point is made by only 2 lines. If 3 or more lines pass through a crossing point, this means that 1 or more regions are collapsed to this point, so we do not have the maximum number of regions.

I drew each shape in PowerPoint. As I drew each line (chord), I would count it, so that I knew how many lines there were, even if the question did not ask for it. Then I opened each image in Paint. I took the area painter (the bucket) and started clicking inside each region, thus counting them. Each region I would click and count was colored by my click, so I knew not to double count it.

And I did not stop there. When I had counted the regions by coloring them with the same color, in the colored shape the lines and the intersection points were still shown. So I took the eraser and I put a little erasing smudge on each intersecting point, thus counting the intersecting points as well, even though the question did not ask for it.

I created the following table which depicts my results. “Points” are the number of points on the circumference. “Chords” are the line segments, where each line segment is drawn between two of the points in the circumference. “Crossings” are the inner crossings, the crossings where two line segments cross inside the circle. “Regions” are the empty areas that are formed inside the circle, “empty” meaning they contain no lines inside them.

The above table shows that the answer to the puzzle’s question is 57.

Let us see how we can generalize our findings. The following analysis is widely known to all mathematicians and also physicists like me know this theory very well. Dare I say, all students know this theory very well? But this measures knowledge or intelligence? I say that it only measures knowledge. Anyway, the test question only asks for the number of regions, so the answer is 57 and that is all. By merely counting the regions we arrived at the answer.

First of all, let us denote the number of points on the circumference with the letter n. This is the column named “Points”.

In order to have the maximum number of regions, each point is connected to each other point with a chord. To find the total number of chords, we can think in one of three ways.

First way: We have n points. Each point is connected with each one of the other n-1 points. So we have n*(n-1) lines (line segments). But in this number, we have accounted for each line twice, from the perspective of both points in each line. So we have n*(n-1)/2 lines.

Second way: From the theory of combinations. The number of lines is

(n choose 2) = n!/[2!*(n-2)!] = n*(n-1)/2.

Third way: We can say that the first point made n-1 connections with the other points and left the building. Then the second point made n-2 connections with the remaining points and left the building. And so on, until the second from last point made 1 connection with the last point at left the building. Thus, the last point is now alone and can make no connection. Thus, the number of connections (lines) made is

(n – 1) + (n – 2) + … + 2 + 1 = n*(n-1)/2.

So, the formula for the column named “Chords” is (n choose 2) = n*(n-1)/2.

Now let us find the formula for the column named “Crossings”.

For 2 lines to create a crossing, the 2 points of one line have to be different than the 2 points of the other line. In other words, to have a crossing, 4 different points on the circumference have to be involved, two of them defining one of the lines and the other two defining the other line.

This concept is easier to grasp if you look at the shape I provide for the simplest case. This is the shape with 4 points on the circumference which corresponds to only 1 crossing. This sole crossing exists because the 4 points on the circumference are 4 different points. Any 4 different points define two lines and 1 crossing.

Thus, the formula for the column named “Crossings” is (n choose 4) = n!/[4!*(n-4)!].

Now let us find the formula for the column named “Regions”, which is what the puzzle describes as regions.

The Wikipedia article I mentioned above rigorously describes the different approaches that explain and provide the corresponding formula. Because someone, like me, may lose themselves in mathematical rigor, before reading the full rigorous explanations and proofs, it is best to have a high level overview of them.

So, as an introduction to these rigorous proofs and explanations, I would like to give a simplistic overview.

Imagine (you don’t have to imagine, you can actually draw what I describe) that we have the circle and nothing else. So, we only have 1 region, which comprises the full circle. Then imagine that we draw parallel lines. As each new line is drawn, another region is created. Now imagine that the lines we already drew start intersecting. As each intersection (crossing) occurs, another region is created.

So, the formula for the column named “Regions” is

1 + Chords + Crossings = 1 + (n choose 2) + (n choose 4).

So, I can present you with the following theoretical table which coincides exactly with the empirical table I gave you above.

]]>

In this blog post, I will study Question 19 from this test.

The question is about giving numerical values to symbols, so that the given equations hold.

Below I present the original question for your convenience:

Please do not read the rest of this article, if you want to attempt to solve this question on your own. The rest of this article describes my attempt at solving this question and you should not read it, unless you want to or you do not mind coming across relevant ideas, spoilers, hints, solutions, and strong opinions concerning this test.

You have been warned and I now consider that you continue to read knowing that what you come across for the rest of this article may forever spoil things for you and/or present strong opinions against this test.

Last warning: please do not read this blog post, unless you are certain that you know what you are doing. If you are not sure, then it would be best if you stopped reading at this point.

OK. If you are here, it means that you want to know my opinion. Well, ok then!

To cut a long story short, my opinion is that the test is highly inappropriate. In this blog post, I will focus on the study of question 19.

This is not a bad question, but I fail to see what it has to do with a test that measures intelligence. It is easy for someone to do some trial and error attempts in order to find the solution. I suppose the test had to had an easy question to tackle. But there are some sinister things going on concerning this question and I am about to bring them forth.

But first, let me provide the answer.

This is a nice question that allows the solver to draw in a piece of paper and try out things and make different attempts. But I do not know how the test was administered. You will shortly understand why I am saying this. This question seeks an answer and it makes clear that the answer is higher than 17. Any attempt, no matter how naive, at solving the problem will not provide a number much bigger than 17. So, by trying different numbers like 18, 19, … the solver is about to find the solution, thus gaming the system.

But since I do not know how the test was actually administered, I will leave this point at that.

But even then, the test expects a number. Giving a number willy nilly, does not say anything. Giving a picture with a path drawn, writing an answer that proves your thinking, posting a blog post like this one, does say something. So, again how was this test administered? I do not know, but I am afraid that the answer might be highly disturbing.

Back to the answer. Below, I provide two images.

First image:

Second image:

From the two images above, we can see that the answer is 19. The minimum numbers of streets to walk on in order to cross all of them is 19. And this is the answer and if you need a little more explaining (do not worry about the word “little”, I am going to provide a whole lot more of explaining in the rest of this blog post) you can observe the two images.

Indeed, you might observe that I provide two equivalent solutions. Two, because there are two images. But wait! In each image, you see that I start from the left and I stop at the right. Or vice versa. The path may be the same but does it count as one or two solutions depending on the direction of travel? I should say that it does not. Your opinion may differ and it is all right. We will discuss more about this later on, anyway.

In both images, the start and stop points are on the left and right middle buildings. But on the first image we observe a point where the path crosses itself. This is like the center of a swastika. When you reach this point, you can go straight or turn 90 degrees. Does this count as two separate solutions or one? Either way, we all know what we are talking about and what we see here concerning the path taken.

But this goes for the second image as well. See that in the same building that is the center of the swastika, the path does not cross itself? I made it so that you can understand the path more clearly. But in reality, it can cross itself, and so different paths and solutions arise, each with 19 steps.

So, in the first image, when you reach the node in the second row and second column from the left, then you can go straight or turn 90 degrees. In the second image, when you reach this node, you can go up or down.

So, the answer is 19, all is ok and we can go and watch Chesapeake Shores (or is it Cheesypeake Shores? Chesapeake Whores? I can never remember!)

And you can stop reading here and all is well. But If you want to go the distance, well, please stay a little longer, dear reader. There are some things that need more explaining and there are some sinister things going on as well.

How am I sure that the minimum is 19?

I will answer, but first let me say something else, that will help me arrive at the answer for the question I just posed.

One of the sinister things concerning this question is that it does not state if we are allowed to choose where to start and where to end our path. Are we allowed? The question supposes that we are, but someone would not know that. But then, how do I know that the question allows that?

I know because this question is about the mathematical subject called Graph Theory and Graphs. And the problem presented here is a common problem in Graph Theory and all common assumptions apply.

And this gives an unfair advantage to someone who knows Graph Theory compared to someone who doesn’t. A huge chasm. A hugely unfair advantage.

Graph Theory makes me certain that the minimum is 19. Someone who does not know Graph Theory would have a hard time to prove that 19 is the minimum, whereas it is easy for me. As I said: Unfair advantage. And I have a problem with unfair advantages. A very big problem.

Anyway, this puzzle pertains to graphs and graphs have their terminology and their common assumptions that make sense in this area of Mathematics.

Here are some links that may help, especially the one about the Seven Bridges of Konigsberg. That Wikipedia article alone is enough to explain what is going on in this question.

To explain something and/or to tell a story, sometimes the most difficult thing is to find where to begin. Our story begins when Euler visited the town of Konigsberg. There, the town folk had a pet peeve. Or a question. Or both. They had healthy inquiring minds. Or not. Anyway, their town is depicted in the map below and the good people of Konigsberg wanted to know if one could go to all four land areas by crossing each of the seven bridges exactly once.

I said that the town folk had healthy inquiring minds, but I am not so sure. To me, it seems obvious that this cannot be done. I cannot find a path that would go to all land areas and traverse each bridge only once. But the town folk either did not have a map like the above (highly improbable), or they were way more inquiring than me and needed a mathematical proof for the possibility or impossibility of such a feat.

Anyway, Euler heard about the problem and thought that this was an interesting problem for him to solve. And solve it he did. He proved that this is impossible. He proved that it is impossible to visit all four land areas by traversing each bridge only once.

How Euler thought and went about solving this problem is crucial for us in order to understand why the answer I gave in the test puzzle is 19 and it is certain that it is 19.

But before I get to that, I must stress that the town folk cared about a path but did not care about which land would be the beginning and which land would be the end of the path. They just wanted a path that would traverse all bridges, each bridge exactly once and visit all land areas.

This notion is an important notion in Graph Theory and is named in honor of Euler as Eulerian path. So, in retrospect, the town folk wanted to know if a Eulerian path exists. And the question of this puzzle is almost the same. It states that there is no Eulerian path and asks us to find the minimum number of bridge traversals we have to make. In the Konigsberg problem, we have land areas and bridges. In this puzzle, we have buildings and streets, respectively.

but the common understanding is that when we are dealing with Eulerian paths, we are not choosy about which is the starting point and which is the finishing point. Of course, in the street cleaner example that the puzzle is all about, let me tell you, the street cleaner would definitely like to have a saying as to where her work would start and where it would end: I guess as close to her home as possible. But the puzzle assumes what is usually assumed in such graph problems. Thus, it is inappropriate for “general consumption”.

But I guess that it would make a great question for a mathematician interested in graphs. Below I will try to explain what is going on in this puzzle, but it will help if you know a few things about graphs already, like studying the links that I provided above.

So, in this puzzle, what we have in front of us is a graph. The nodes (or vertices) of the graph are the buildings. The edges of the graph are the streets. This graph is an undirected connected graph. “Undirected” means that the edges have no direction (for each edge there are two nodes, but not a beginning and an ending node, just two nodes). “Connected” means that no “islands” exist. There is always a path from a node to another node, but it might be through another node. This is why I did not characterize the graph as “complete”. It is not “complete”. Not all possible edges exist. This graph is just “connected”.

So we have an undirected connected graph and we want to study the Eulerian paths. Actually, we know that there are no Eulerian paths, but we want to find the minimum next best path.

Before we get to that, I would just like to bring your attention to the difference between Hamiltonian and Eulerian paths. A Hamiltonian path is one that visits each node once. A Eulerian path is one that visits each edge once. And in both, it does not matter where we begin and where we end.

When Euler studied the Seven Bridges of Konigsberg, he understood a few very important concepts. He understood that what plays the defining role is the number of bridges (edges) that each land (node) has attached. The number of edges that are connected to a node is called the degree of the node. And it is the most important thing in order to study the existence of the Eulerian paths.

You see, Euler understood that during a walk, the path you go along traversing land areas and bridges, will bring you in and out of land areas. Let us exclude the beginning and ending land area for a moment. All other land areas will be visited by going in one bridge and leaving in another (a different one, a different bridge) since we do not want to traverse the same bridge twice. And so he realized that for a Eulerian path to exist, all intermediate nodes in the path have to have an even degree, that is, an even number of edges connected to them. Now for the beginning and ending node, they have to be both of either odd or even degree. If one of them is of odd degree and the other one is of even degree, then a Eulerian path does not exist. If there are two nodes of odd degree, then anyone of them will be the beginning node and then the other will have to be the ending node of the Eulerian path. If there are zero nodes of odd degree, then a Eulerian path definitely exists.

So, for a Eulerian path to exist, all nodes have to be of even degree, or two nodes have to be of odd degree, In the latter case, each one of them can be the beginning node of the path, but once we chose the beginning node, the other node has to be the ending node of the path.

Now we have all the concepts we need to solve the puzzle in the most mathematical and precise way there is.

The Seven Bridges of Konigsberg did not have a Eulerian path because the corresponding graph contained 4 nodes and each one of those had an odd degree.As you can see in the map, three land areas had three bridges connected to them and one land area had five bridges connected to them. So, all land areas had an odd number of bridges connected to them. I wish I could go to Konigsberg and scream to the ear of each town folk: “Hey baster, there is no Eulerian path, because in this connected graph, there are four nodes and all have an odd degree. Get it? No? You need to have zero or two nodes of odd degree in order to have a Eulerian path! Get it now? You stupid %@^&!”.

The same goes for the corresponding graph for this puzzle. There is no Eulerian path, because there are six nodes of odd degree. The following image depicts these six nodes by having circles around them. I circled them and I also used the green color for the nodes on the far left and far right and the red color for the nodes in the upper an lower part. I did this in order to facilitate the discussion that is going to follow.

In the solutions that I gave, I always start and finish at the green nodes. And I always traverse twice the two pairs of red nodes, the pair in the upper part and the pair in the lower part. This is not coincidental. By traversing the red nodes twice, it is as if I create another edge among them. It is as if I create another street. Please note that in the two images that depict the solutions that I gave, the pairs of the nodes have another red line going backwards and in a slight angle. This means that I traversed this path twice. And it can equivalently mean that I created another street between these two buildings. And by doing so, I changed the degree of the nodes (buildings) from odd to even.

Thus, it only takes one more edge between the two upper middle nodes and one more edge between the lower middle modes to turn this graph in to one with two nodes of odd degree (the nodes circled with green color). And this is why all solutions start and end in these two green nodes. Because, by traversing twice the red nodes edge, it is as if I created another edge, thus making them into even degree nodes, thus leaving the green circled nodes to be the only two odd degree nodes, so they had to be the beginning and ending node of the Eulerian path that was just created by the virtual addition of the two new nodes that my double traversing made.

Now, one may ask if we can have another pair of the six odd degree nodes as the beginning and ending node. The answer is that we cannot. Actually we can create an edge between any two of them and another edge between any other two of them, thus leaving again only two nodes with an odd degree. But the edges that we will draw will not be 1 kilometer long. There will be lengthier. So, although we can create a Eulerian path by leaving any two odd degree nodes, the only case that we can do so and with the minimum distance is if we connect the adjacent pairs of the red nodes.

And this is how I know that the solutions that I gave are optimal, that the beginning and ending building are those on the middle far left and right and that the minimum path needed to traverse all streets is 19 kilometers long.

]]>In this blog post, I will study Question 2 from this test.

The question is about giving numerical values to symbols, so that the given equations hold.

Below I present the original question for your convenience:

Please do not read the rest of this article, if you want to attempt to solve this question on your own. The rest of this article describes my attempt at solving this question and you should not read it, unless you want to or you do not mind coming across relevant ideas, spoilers, hints, solutions, and strong opinions concerning this test.

You have been warned and I now consider that you continue to read knowing that what you come across for the rest of this article may forever spoil things for you and/or present strong opinions against this test.

Last warning: please do not read this blog post, unless you are certain that you know what you are doing. If you are not sure, then it would be best if you stopped reading at this point.

OK. If you are here, it means that you want to know my opinion. Well, ok then!

To cut a long story short, my opinion is that the test is highly inappropriate. In this blog post, I will focus on the study of question 2.

This question is one of the most inappropriate questions on this test. Let me begin my analysis, where it will become very clear why I have this bad opinion about this question.

I will arrange the equations given as follows and I will begin to solve them in that order:

When I began solving this puzzle, I was skeptical. Part of me said that this would be a question that would be solvable with reasonable assumptions and part of me said that this would be a question that needed unreasonable guesses. In hindsight, I cannot believe how right my latter part was and actually this is also an understatement.

I began with trying to guess the first three equations, i.e. those that had numerical values on their right side. Since 23 and 29 are prime numbers, I assumed that symbols next to each other were simply added (as opposed to multiplied), and, in hindsight, I was correct.

I tried to give values to the symbols in the first equation, so that the second equation would hold.

So, I came up with the following for the first equation:

2 + 3 + 3 + 3 + 3 + 4 = 18 or 4 + 3 + 3 + 3 + 3 + 2 = 18

so that the components of the second equation would be like:

3, 2^4=16 or 4^2=16, 2*2=4 or 2^2=4

and the second equation would be 3 + 16 + 4 = 23.

Things were looking up and I had options. Unfortunately, none of my options seemed to satisfy the third equation. I struggled a lot, until I discovered that the only way to satisfy all first three equations was to have GreenCircle = 2, RedSquare = 3, YellowTriangle = 4 and, and here is the kicker, BaseYellowSquare representing the raise to the second power.

So, when the yellow triangle is alone as a symbol, it is equal to 4. When it is under another symbol, it raises the value of the symbol to the power of 2. And when a yellow square is underneath another yellow square, the yellow square which is above represents the number 4, whereas the yellow square which is underneath raises said 4 to the second power.

For the yellow square to have two different interpretations, depending on whether it was alone (or on the very top of a structure) and whether it was used as base with a symbol on top, is absurd. But since it was the only thing that could make the first three equations working, I went along with it, half-heartedly and with doubt, because of Occam’s razor. “Surely there must be a more sensible interpretation”, I was thinking to myself. In hindsight, things were to become way more absurd, to the final point of complete absurdity. You will see why later on.

So, at this point, things look as follows:

First equation: 2 + 3 + 3 + 3 + 3 + 4 = 18.

Second equation: 3 + 2^2 + 4^2 = 23.

Third equation: 3^2 + 4^2 + 2 + 2 = 29.

From this point, the fourth and fifth equations can be studied independently. From the fourth equation we can get the value of BlueRhombus and from the fifth equation we can understand what the BaseGreenCircle represents. Because, if we are to learn something from our previous experience, symbols here, when used as a base, represent operations rather than numbers.

Let us continue by studying the fourth equation. Let x denote the value of BlueRhombus. We have:

x^2 + (x+3)^2 + 3 = (x+4)^2 + 3^2 + 2 =>

=> x^2 + x^2 + 2*3*x + 3^2 + 3 = x^2 + 2*4*x + 4^2 + 9 + 2 =>

=> x^2 + 6*x + 12 = 8*x + 27 =>

=> x^2 – 2*x – 15 = 0 =>

=> x = -3 or x = 5, and I keep the positive value x = 5.

Thus BlueRhombus = 5. Nothing absurd here. We are at a very sensible point.

Let us continue by studying the fifth equation. The right side is equal to:

3^2 + 4 + 4 + 2 = 19. Now let us study the left side. There is a green circle with 3 red squares on top and a yellow triangle with another yellow triangle on top and on top of the second yellow triangle there is a green circle.

Obviously, the yellow triangle structure is (2^2)^2 = 4^2 = 16. This is because the triangle that is on top along with the circle that is on top of it amount to 2^2 = 4. So the bottom triangle contains the value 4, thus it amounts to 4^2 = 16.

So, if the triangle composite is 16, this leaves the green square structure to be 3. But the 3 red squares that are on top are equal to 3 + 3 + 3 = 9. So, having a green circle as base corresponds to calculating the square root. The square root of 9 is 3 and the green circle with the 3 red squares is equal to 3, because the 3 red squares amount to 9 and 3 is the square root of nine.

Thus, BaseGreenCircle corresponds to the square root calculation.

So, the absurdity continues. When a green circle is used alone (or on the very top of a structure) it is a 2 but it is used as a base it is the square root calculation.

What I found most absurd in the above was that the square root calculation had entered the picture. I was unsure and uneasy with my interpretation, since I believed that the test was about judging intelligence and not mathematical knowledge. And I consider the square root to be a more advanced concept that multiplication or raising to a power. But all evidence pointed to the fact that indeed the BaseGreenSquare was the operation of the square root calculation.

So, I accepted all the above with some uncertainty and skepticism. Little did I know that this was just the tip of the iceberg.

So, we are the following point:

It seems that we are close to the end, and indeed we are, but we are in for a big surprise. Before we proceed, let us recapitulate:

GreenCircle=2.

RedSquare=3.

YellowTriangle=4.

BlueRhombus=5.

BaseYellowTriangle produces the power of 2 of whatever is on top of it.

BaseGreenSquare produces the square root of whatever is on top of it.

I could not find anything more simple to interpret the first five equations, so I was content that I was near the end. I just had to interpret the sixth and seventh equation and then I would have to calculate the eight expression. To interpret the sixth and seventh equation meant to interpret what the BaseBlueRhombus meant, i.e. what operation the blue rhombus performed when it was used as a base for something.

This is because both the sixth and the seventh equation feature the blue rhombus as a base in their left side.

Let us study the sixth equation. The second part is 5. The first part is the blue rhombus as the base and a blue rhombus on top. So we need to find what operation, let us call this operation f, the blue rhombus denotes. From the sixth equation we have that this operation transforms a 5 to a 5.

Let us now study the seventh equation. The right part is equal to:

4^2 + SQRT(4) + 3 = 16 +2 + 3 = 21.

The left part is a blue rhombus with two yellow triangles on top. Tow yellow triangles are equal to 4 + 4 =8. So, from the seventh operation we have that a blue rhombus transforms an 8 to 21.

Let us recapitulate. We are trying to find the mathematical operation f that BaseBlueRhomus corresponds to. And we have that: f(5) = 5 and f(8) = 21. Once we know what f is, it will be straightforward to calculate the eighth and final expression. This is because the left side of the eighth equation is f(f5+2)) = f(f(7)).

This is easy to see. The left side of the eighth equation is a blue rhombus with another blue rhombus on top and the second blue rhombus has a blue rhombus and a green circle on top. The blue rhombus and the green circle on top are equal to 5 + 2 = 7. So the blue rhombus that is on top is equal to f(7) and the whole structure is equal to f(f(7)).

So, we are the following point:

So, what is f? What is this mathematical operation that sends a 5 to a 5 and an 8 to a 21? From the moment I encountered this puzzle, I thought of the Fibonacci numbers. (Here indeed, 5 and 21 belong to the Fibonacci number sequence.) But the moment I thought of the Fibonacci numbers, I dismissed them. Here in Greece, this is an advanced concept that students do not learn as part of their mathematical education. Here in Greece, I guess that concepts such as the Golden Ratio, Fibonacci numbers and the squaring of the circle are considered esoteric, advanced, unscientific and they are not taught as part of the curriculum. So, my instinct was to avoid Fibonacci numbers, because they are way too advanced for a puzzle that measures intelligence by balancing equations.

Let me explain this a little bit further, so there are no misunderstandings. I am the only one that thinks pi, the Fibonacci number sequence, numbers in general are mystical, esoteric entities. Scientists scoff at notions like mine. Although I treat the mathematical concepts as divine, all others have completely demystified them. I agree with their interpretations, although I think there is something more hidden underneath all our understanding. Anyway, although I find concepts such as those to be esoteric, I vouch for them. This is in contrast with main science which finds these concepts to either be non-esoteric. Because the Fibonacci sequence is a concept that some people like me believe to be esoteric, the Greek curriculum avoids it. Or this is what I think that the reason is.

So, I ignored the Fibonacci number sequence and all other esoteric mathematical entities and I tried to find a mathematical operation that turned a 5 into a 5 and an 8 into a 21.

I thought about decimals, fractions, factorials, but I sort of focused with modulo arithmetic and mostly division. What if this strange elusive operation f was modulo division 5? After all, 21 = 2*8+5, thus 21 mod 2 = 5. I tried desperately to make a 5 go to a 5 and an 8 go to 21 with all sorts of modulo operations, but I couldn’t find any that would persuade me that this is what the authors had in mind.

Occam’s razor. Again. Always.

So, I am back to the Fibonacci sequence. What was Sherlock Holmes saying? “Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth.” Then, that is that. I made sure to eliminate all other mathematical operations. But I was still uneasy. The Fibonacci sequence is not an operation. It is a number sequence.

Let me give you the final part of the solution and here you will understand why I am still uneasy.

So, the Fibonacci sequence is produced by starting with 0 and 1 and from then on each number in the sequence is the sum of the previous numbers in the sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, …

If we assign an index of 0 to the first element, then for the function f that supplies a number for the sequence, we have:

f(0)=0, f(1)=1, f(2)=1, f(3)=2, f(4)=3, f(5)=5, f(6)=8, f(7)=13, f(8)=21, f(9)=34, f(10)=55, f(11)=89, f(12)=144, f(13)=233, f(14)=377, f(15)=610, f(16)=987, f(17)=1597, f(18)=2584, f(19)=4181, f(20)=6765, …

So, from the above, we have that:

f(5)=5, thus the sixth equation holds

and

f(8)=21, thus the seventh equation holds.

So let us compute the eight expression:

f(f(5+2))=f(f(7))=f(13)=233.

Thus, the answer is 233.

Now, what makes me uneasy is that the Fibonacci sequence is not an operation. OK, I admit that if we want to stress things, an operation can be seen as a function and here we have a function f that returns a number from the Fibonacci sequence.

But this is a puzzle concerning the balancing of symbol equations. Only a crazy person would assign the symbol of a blue rhombus to be the function of the Fibonacci sequence and the symbol that is on top of it to be the index of the Fibonacci sequence. You see, I have seen many puzzles concerning the balancing of symbol equations and to me this is crazy thinking.

This leads me to another point I would like to make. I believe that this test is highly inappropriate, because it wants you to be knowledgeable and it wants you to guess. The latter is way worse than the first. The mentality of this test is: “Guess what I am thinking and we will see if you are lucky enough”. And also “I know of this mathematical theory ad discovery and paper that if you too have happened to come across, then you will be lucky enough to be able to answer this question”.

The whole morality of this text is off. Also, I found things there that made me question my sanity.

So, here what I will assume. I assume that this test was put forth to see who objects. I imagine the authors being somewhere shaking their heads in contempt. “No one opposed it!” they will cry in dismay. “No one had the decency to object! Where has this world gone to! Only one person had the integrity to question us. And this person is Dimitrios Kalemis.”

Of course, I do not really believe the above, that this test was put forth to fool us and test us, but it would be the only sensible explanation for me.

I believe that the authors wanted to find highly intelligent individuals to help each other towards a good purpose. I do not know. But this is what I believe. The problem is that the authors claimed to have created a test that measures high intelligence, whereas I disagree.

But the main issue is the following and it reminds me of the P vs NP problem:

Suppose I pose a question that you understand. And suppose that you devote time and energy to try to answer this question. And suppose that you fail to answer this question. And suppose that the answer to this question is an answer than when I give it to you, you will understand it. Then, I say that it is my moral obligation to give you the answer.

The authors of the test do not provide the answers to this test.

If the answers to this test can be understood, then my claim is that the authors have the moral obligation to provide them.

Since the answer of a question is not available, and I happen to know it, I have the moral obligation to provide it.

I am scientist and scientists publish the results of their research. And I just published here my research and these are my results. And this is how the question is answered and this is what it means for the validity of the test.

]]>In this blog post, I will study Question 21 from this test.

The question is about the construction of a square from 24 non-overlapping smaller squares, each with a side of unique length.

Below I present the original question for your convenience:

OK. If you are here, it means that you want to know my opinion. Well, ok then!

To cut a long story short, my opinion is that the test is highly inappropriate. In this blog post, I will focus on the study of question 21.

Actually, this one of the good questions of the exam. I am saying this because the exam contains some highly inappropriate questions as well. But there is still a problem with this question. As with some other questions in this test, the answer can be found in one of Martin Gardner’s books.

So, why would Haselbauer and Dickheiser copy (not just get inspiration, but blatantly copy) questions from Gardner’s books, questions that are fully answered in those same books, when they claim, and I believe them, that they want the integrity of the test to be high?

I said and I am planning to prove that the Haselbauer-Dickheiser Test is inappropriate (and this for many reasons), but this is ridiculous!

Anyway, back to answering the question. I will provide information about Martin Gardner’s book later on in this blog post. But first, I would like to present my attempt at solving this question, which I did before I read the corresponding chapter of the book.

So here is how I solved this question without reading the corresponding chapter.

I observed the image and I thought that I should tackle the problem beginning from the smallest square. I named each of the 24 squares with a number from 1 to 14, counting in my order of preference that I chose loosely based on the order in which I imagined that my calculations would proceed. Anyway, giving a name to each square is arbitrary, because the actual name does not matter, what matters is that we need to have a way to refer to each square. So, I chose the names that are depicted in the following image:

Please note that image above does not depict the size of the side of each square. It depicts the name of each square. I made this so that I can refer to each square and so that I can calculate the size of the side of each square.

So, imagine that the smallest square is named 1, or rather A1, because I am going to use Excel and this way, I will easily create the spreadsheet so that the size of each square will correspond to the cell that has the name of that square.

Here is the Excel spreadsheet that I created.

And here is the way I worked, what I was seeing when I was working at the solution:

So, I had these two windows side-by-side as I was working with the spreadsheet.

Let me describe the spreadsheet. Next to each cell is another cell that depicts the formula used in the first cell. I only enter numbers in the four white cells. The numbers in the yellow cells are automatically calculated by the corresponding formulas in those cells.

Cell A1 corresponds to the side of square 1, the smallest square. Cell A2 corresponds to the side of square 2, the one on the left of square 1. I do a trial and error analysis, where I guess the values of the sides of squares 1, 2, 14, and 15. And I let the formulas that I entered produce the values of the sides of the other squares.

So, the side of square 3 is on the cell A3, which contains the formula =A1+A2, because the side of square 3 is the sum of the sizes of squares 1 and 2. And so on for the other squares.

So, by guessing the sizes of four squares, I calculate the sizes of the rest of the squares. The values I choose have to be small for the squares 1, 2 and 14, and the value for the square 1 has to be the smallest of the three. Also, by looking at the value that is produced from square 1 and square 2 for square 10, I choose a somewhat smaller value for square 15. This is because the image gives me this hint, if I roughly compare square 15 and square 10.

In order to know if the values I enter by trial and error produce the correct result, I check that the sizes of the all encompassing square are all equal. This is why I have created the column D. Each cell is the resulting calculation of one of the four sizes of the all encompassing square. By entering values in the four white cells in column A, I do not really care about the values produced in the yellow squares of column A. I really care about the four values in column D. They all have to be equal to each other. So this is how I know if I have the correct values in the four white cells in column A.

By trial and error, I found that when A1=1, A2=3, A14=2, and A15=29, all values in column D match, i.e. they are equal to each other and equal to 175. So, this means that I solved the puzzle. So, the all encompassing square has a side equal to 175. And the square in question, square A10, has a side equal to 38. And this is the answer: 38.

The way I solved the puzzle, I was very cautious that the numbers I input might have been coprime, especially A1 and A2. When I first saw this puzzle and understood that I would solve it using trial and error, basing my guesses and beginning from the smallest square and the one next to it, I was extra careful not to assume any relationship between these two numbers. In hindsight, things were not that bad, since A1=1 and A2=3, but I could not have known. So I was extra careful to account for a case like A1=2, A2=5 and so on.

Another thing is that the question should have been stated in a way that acknowledges that there are infinite such rectangles and we are searching for the smallest one. Indeed, any integer multiple of the solution is a solution as well. For example, multiplying by 2, we get A1=2, A2=6, A14=4, A15= 58 which also produces a solution. And multiplying by 3, we get A1=3, A2=9, A14=6, A15= 87 which also produces a solution. And so on to infinity. But the question forgets, or rather assumes that we correctly assume it. Good assumption from our part, bad assumption from the question’s part.

As I said, the answer is depicted in one of Martin Gardner’s books. Martin Gardner’s book titled “The Second Scientific American Book of Mathematical Puzzles and Diversions”, which is the 2nd book out of a 15-book series has the answer to this puzzle. Below, I present a screenshot of the pages 205 and 206 from this book.

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In this blog post, I will study Question 16 from this test.

The question is about the maximum number of pieces a donut (doughnut) can be sliced with three simultaneous plane cuts. Well, the “maximum” part is implied but not stated clearly. And by “donut”, they mean “torus”.

Hmmm… some question! Ask any policeman, they’ll know.

Below I present the original question for your convenience:

OK. If you are here, it means that you want to know my opinion. Well, ok then!

To cut a long story short, my opinion is that the test is highly inappropriate. In this blog post, I will focus on the study of question 16.

Actually, this is a good question and tests your ability to think in 3D. If you are so inclined, you could do the imaging in your head. In this case, I would suggest to imaging the torus being very big. This will help, because there are some small pieces produced as well.

Or you could use your favorite 3D modelling software to draw a torus and three planes cutting it. You should choose a 3D modelling program that not only is your favorite bit that it also allows you to view everything in 3D and rotate and translate the model in real time. This is because you will want to see your model from many different angles.

I would also recommend that you alternate between making the objects (the torus and the three planes) transparent, semi-transparent and opaque, in order to design and study your model better.

I like POV-Ray, but for this purpose I would strongly recommend something along the lines of Blender. Anyway, if you use POV-Ray, here is how you could create a torus:

torus { 1, 0.4 rotate < 90 , 0, 0> pigment { rgbt <0,1,0,0.0> } }

and here is how you could create a plane:

polygon { 4, <2, -2>, <2, 2>, <-2, 2>, <-2, -2> rotate < 0 , 60, -20> translate <2,0,0> pigment { rgbt <1,0,0,0.0> } }

With the last decimal in the pigment rgbt you can control the opacity of the object.

OK. Now, let;’s get to the answer. A few paragraphs above I suggested that you ask a policeman. That was, of course, a joke. Who wants to address a pig? But, when it comes to a cool guy like Martin Gardner, well I would certainly want his opinion.

Turns out, Martin Gardner’s book titled “The Second Scientific American Book of Mathematical Puzzles and Diversions”, which is the 2nd book out of a 15-book series has the answer to this puzzle. Crazy, huh? Well things are about to get way more crazy and ridiculous. This same book contains the answer for another of the Haselbauer-Dickheiser test’s questions! And other books in the series contain the answers to some other of the test’s questions.

So, why would Haselbauer and Dickheiser copy (not just get inspiration, but blatantly copy) questions from Gardner’s books, questions that are fully answered in those same books, when they claim, and I believe them, that they want the integrity of the test to be high?

I said and I am planning to prove that the Haselbauer-Dickheiser Test is inappropriate (and this for many reasons), but this is ridiculous!

Anyway, back to answering the question. I will not answer, I will let *the* man Martin Gardner to do it. He is way to the infinity better than me.

Below, I took a screenshot of pages 149 and 150 from Martin Gardner’s book.

The answer is that we can create as many as 13 pieces, albeit some of them very small compared to the others.

Another, more refined picture of the above, which I found online:

and another:

Let me help you better understand what is going on. The reason that we are able to create an odd number of pieces is because of the somewhat vertical plane not being in “alignment”. In the image below, I highlighted two intersections and you should note that the blue line is not parallel to the red line.

Anyway, rather than looking at still images, it is better to create a 3D model and view it from all perspectives.

So, the answer is 13 pieces.

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In this blog post, I will study Question 24 from this test.

The question is about a probability calculation concerning two coins, one of which is heads on both sides.

Below I present the original question for your convenience:

OK. If you are here, it means that you want to know my opinion. Well, ok then!

To cut a long story short, my opinion is that the test is highly inappropriate. In this blog post, I will focus on the study of question 24.

This question is highly inappropriate as well, as are most (if not all) of the questions on this test. The reason this question is inappropriate is that it does not test cleverness but knowledge. If you have been taught probability theory and the concept of sample space, then you can find the solution. Otherwise, you are clueless. Probability theory and the concept of sample space is one of the domains that the following brilliant comic is right on the spot.

So, we have two coins. One coin is heads on both sides (um…. who comes up with these scenarios?). The other coin is heads on one side and tails on the other side. Ok, that’s the normal coin. The first coin is the abnormal one, the “rebel”.

If the question just stated that one coin is selected at random and we examine one of its faces at random, then we would have the following 4 cases (the following sample space):

Coin H-H is selected and its first face H is examined.

Coin H-H is selected and its second face H is examined.

Coin H-T is selected and its first face H is examined.

Coin H-T is selected and its second face T is examined.

But the question states something more: That the face that we examine is heads. So the sample space ends up having the following 3 cases:

Coin H-H is selected and its first face H is examined.

Coin H-H is selected and its second face H is examined.

Coin H-T is selected and its first face H is examined.

So, only the 3 cases above could have happened.

Now, from the 3 above cases, only the first 2 correspond to the other face being heads.

Thus, the probability we are looking for is 2/3.

]]>In this blog post, I will study Question 13 from this test.

The question informs us that we have eight identical squares in size and we have to find the order in which the y were place to provide the configuration depicted.

Below I present the original question for your convenience:

OK. If you are here, it means that you want to know my opinion. Well, ok then!

To cut a long story short, my opinion is that the test is highly inappropriate. In this blog post, I will focus on the study of question 13.

Actually, I find that this is the least obnoxious of the question. Still, I do not consider the question fit to examine a person’s intelligence. But it is not completely hopeless, either, unlike the rest of the question on this test.

When I encountered this question, I thought that the solver did not need to make any assumptions. So, this is what is right about this question.

So, I observed and observed and observed the configuration. And since I did not come up with anything conclusive at first, I thought about importing the picture in PowerPoint and drawing the vertical and horizontal lines that you see in the following image.

And I immediately understood what was going on and the order that the squares were placed. And I wrote the order from bottom to top at the comments section at the bottom of PowerPoint, as you can see in the image above.

So, the question asks us to write the order from top to bottom, but it helps us to understand how the squares were placed by going from bottom to top. Then it is straightforward to reverse the order.

So, I solved the question, but I wanted to make absolutely sure and to prove that was the answer. So, I used Excel. And I drew the squares one by one as they formed the configuration. My work using Excel is depicted in the following image.

So, the order from top to bottom is A D G H F B E C.

]]>In this blog post, I will study Question 22 from this test.

The question gives a 10 X 10 square grid. Each square has a color and a single digit number. Also, a number appears after each row and each column, except after the third row, where there is a question mark. The question makes obvious that we have to find the number that goes in the place of the question mark.

Below I present the original question for your convenience:

OK. If you are here, it means that you want to know my opinion. Well, ok then!

To cut a long story short, my opinion is that the test is highly inappropriate. In this blog post, I will focus on the study of question 22.

When I first encountered the question, I saw the numbers outside the grid and they looked like sums to me. They looked like a the result of a calculation that involved the corresponding row or column. In hindsight, yes, I was correct. But the question does not state this. You have to guess. Guessing is not a good thing. Guessing is a highly inappropriate thing. I am talking in the context of an IQ test. It is so easy for me to create an IQ test like that. I will name it “Guess What I Am Thinking!”. And you will have to guess whatever rule I coined. That’s not right.

So, as it turns out, these numbers are sums. But if they are sums, is the question ill-conceived? And does the question allows you to cheat?

What I mean is that if these numbers are sums and each square is summed depending on its color and its number, then the sums of the column sums will be the same as the sum of the row sums.

In other words, we should have:

95 + 93 + 62 + 77 + 106 + 100 + 102 + 100 + 78+ 97 =

= 80 + 92 + x + 89 + 98 + 91 + 78 + 99 + 83 + 88

And from the above equation, we have x = 112.

Indeed, in hindsight, this is the correct answer. In this blog post I will explain why I am certain that this is the correct answer.

But I do not think that the examiner wanted the solution to be found this way. Again I am guessing. There is and will be a lot of guessing involved in this test. And this is one of the things that is bad about it.

So the correct answer to life, the universe and everything, or just the missing number in this question is 112. But if the test was administered in an environment where you could type your answer and receive the result of whether your answer was true or false, you could easily game the system and try 112 or other different numbers and then, when you would have been informed that your answer was correct, you could use this information to trace things backwards.

So, one of the problems pertains to the way the test is administered. Answering 112 does not amount to much. You have to state why you gave the answer that you gave. But I am afraid that the test is not administered in this way, but it is administered in a way that allows you to cheat and game the system.

OK. Let us suppose that we are not certain that the numbers outside the grid are sums and we are not certain that the sum of the numbers in the bottom is equal to the sum of the numbers on the right. How will we proceed to answer this question?

Here is what we have to do. We will assume that each number outside the grid is the sum of the numbers in the corresponding row or column. But each number will have to be differentiated according to the color of the square that it resides on.

Any normal person would assume that each color corresponds to a number, and in hindsight, this is correct. Any normal person would also assume that the number of the color multiplies the number of the digit. But in hindsight, this is not correct.

Anyway, I dis not know this and I assumed that the number corresponding to the color in each square multiplied the digit in the square. So I had to find the number that corresponded to each color.

Here is what I did. I observed and observed and observed the grid. And I noticed that the sixth column from the left contains only four colors (pink, green, blue, red) of the five colors in the grid (pink, green, blue, red, yellow). Specifically, in the sixth column we have the following:

The sum of the red squares is 9 + 2 + 6 + 4 = 21.

The sum of the pink squares is 5 + 3 = 8.

The sum of the green squares is 4.

The sum of the blue squares is 4 + 8 + 4 = 16.

And the sum (I loosely refer to it as sum, it is really the result of an unknown calculation) of the whole column is 100.

Since I was assuming that each color multiplies the digit by a constant number, I immediately knew that the red color corresponded to an even number. Why? Because the sum was an even number (100) and the sums from the other colors each was an even number. So, even if their color corresponded to an odd number, the multiplication would end up in an even number. Since the whole sum had to be even, it was without doubt that the red color was an even number.

So, red had to be 2 or 4, because had it been 6 or higher, 21 * red would be more than 100. Actually, red could only be 2, because if red was 4, then even if the other colors were each 1, the sum would end up more than 100. Also, each color had to be a different number, because it would be illogical not to.

So, it should be that red =2 and pink, green and blue had to have different values.

The problem was that the equation

21 * red + 8 * pink + 4 * green + 16 * blue = 100 =>

=> 21 * 2 + 8 * pink + 4 * green + 16 * blue = 100 =>

=> 42 + 8 * pink + 4 * green + 16 * blue = 100 =>

=> 8 * pink + 4 * green + 16 * blue = 58

does not hold for any integer values we may try for pink, green and blue, given that we have to choose from the integers 1,3,4,… and all three should be different. (I remind you that 2 is missing, since red =2).

So, I struggled with this for a while, until I gave up. So, it seemed that the multiplication concept was not what was going on in the calculation. So, how where the “sums” produced? I thought that if there was not a multiplication of the color number, it would be an addition of the color number.

So, I assumed that each color corresponded to a different integer and that integer was added to the digit that was in the square. And, in hindsight, that was the correct assumption.

So, I had to test my assumption. I had to assign a different integer number to each one of the five colors and calculate the sum of each column and row, where for each square I would add the color number to the digit.

But I had to do a what-if analysis. I wanted to try different values for each color to get the sums to match those given. So, here is what I did: I used Excel.

I created five names, one for each color, and I would set integers as their values, to see if the sums would match those that the question gave.

Each cell was the sum off the digit and the color. The digit was a number and the color was an Excel name. So, the cell A1 would have the formula =3+red. The cell B1 would have the formula =4+green. And so on.

In hindsight, it would have been easier for me to do all the checks I made, if I did not use names but instead I would use cells. This is because I had to open name manager each time I wanted to make a change, whereas a cell would always remain open in front of me, thus making any changes and what-if analysis easier.

Anyway, I struggled a little bit, trying different values, always being careful to use different values for different colors and doing the analysis primarily on columns which contained fewer than the five colors. Whenever I got a sum that was correct in one row or column, another sum would be off, so I struggled a little bit, I tried different permutations of integers and I finally got one that produced all sums correctly. The is depicted in the following screenshot.

I am certain that this is the correct answer, since it would be highly improbable for the examiner to have something else in mind. Each row and column sum is validated and checked to be in agreement with what the question provides.

So now we know that the numbers outside the grid are the sums of each row and column, where by sum we mean the calculation that takes each square and adds the color to the digit and then adds all these sums. Blue = 7, Green = 2, Pink = 6, Red = 4, Yellow = 3. The missing number is 112.

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There are two players that alternate playing.

Each player HAS to take A LEAST ONE object up to AS MANY AS they like including ALL objects BUT ONLY FROM ONE OF THE ROWS. Which row this should be is up to the player to decide.

The two players decide in advance who wins and who loses. Either they decide that the player who picks the last object wins or that the player who picks the last object loses.

I will now present an analysis of the game.

I will start by giving an example configuration.

Suppose we have the following 4 rows of objects (coins, paperclips, cards, it doesn’t matter):

object

object object object

object object object object object

object object object object object object object

Let me transform the above to an equivalent representation that depicts the number of objects in each row:

1

3

5

7

Let me transform the above to an equivalent representation that depicts the number of objects in each row in binary number format:

001

011

101

111

We are interested in the 1’s. In the above configuration, the 1’s are balanced.

What does it mean that the 1’s are balanced? It means that each column contains an even number of 1’s.

So, we call this configuration balanced.

Well, the state of the game is completely determined at this point. Completely determined.

Not only is the state of the game completely determined at this point, but it is completely determined at any point, even if we begin with more or less rows and/or more or less objects.

The state of this game is completely determined if we have five rows of 363, 356, 5467, 4674, 254 objects, or six rows of 2, 4, 5, 35, 1000 objects or whatever else we can think of.

I will explain why I am saying this, but first I would like to explain what I mean by completely determined. I mean that both players know the ultimate winning strategy and no player plays in a naive manner.

If these conditions are met, the state of the game is completely determined before it even begins, if we are given the following:

1) The number of rows and how many objects each row has (the whole configuration).

2) Who is going to play first and who is going to play second.

3) Who wins: the player that takes the last card or the player who does not.

Given the above, the state of the game is always predetermined. (Actually, as we will see, number 3 is irrelevant. The first two conditions fully determine the outcome of the game.)

Given conditions 1 and 2, the state of the game is always predetermined and this is because a configuration can always be transformed to an equivalent representation that depicts the number of objects in each row in binary number format.

And from that representation we can find if the configuration is balanced or unbalanced.

If a player has a balanced configuration in front of her and she has to play, she will create an unbalanced configuation. Always. Even if she wants to or not.

If a player has a unbalanced configuration in front of her and she has to play AND SHE KNOWS THE ULTIMATE WINNING STRATEGY, she will create a balanced configuration if she wants to. Always. If she wants to.

So, between two players that know the ultimate winning strategy, the game will be predetermined.

So, who will win? I will tell you who will lose. The player that first is in front of a balanced configuration and is her turn to play, she will definetely lose.

So, if we begin with an unbalanced configuration, the first player will choose to create a balanced configuration. So the second player will have in front of her a balanced configuration, thus she will definitely lose. I will explain why later on.

And if we begin with a balanced configuration, the first player will definitely lose. For the same reason as before, which I will explain later on.

Before I give you an example of the ultimate winning strategy, I would like to give you an example of unbalancing and balancing.

Suppose we are at the beginning:

1 001

3 011

5 101

7 111

The above is balanced, because each column has an even number of 1’s.

Player1 will have to play because she is the first player.

No mattter what she does, she will leave an unbalanced configuation for Player2.

Suppose Player1 takes the whole last row. Player2 is left with an unbalanced configuration:

1 001

3 011

5 101

Wow! This is hugely unbalanced. All three columns have an odd number of 1’s. This is as unbalanced as it can be!

Player2 can, if she wants to, go back to a balanced configuaration. How? Here is how: She can take 3 objects from the last row, thus creating the following balanced configuration:

1 001

3 011

2 010

Ok, now let us see who wins and who loses.

I will consider the configuration given at the start of this analysis.

I assume that both players are knowledgeable and do not play in a naive manner.

The state of the game is always and completely determined when it is the turn of a player to play and this player is in front of a balanced configuration.

When a player is in front of a balanced configuration and it is her turn to play, then she will definitely lose the game. Definitely. Always. Assuming of course that the other player is knowledgeable. So she will always lose the game, even if she is knowledgeable, too.

When a player is in front of a balanced configuration and it is her turn to play, the fate of the game is completely determined and she will definitely lose, always, no matter if the player who takes the last object wins or loses. No matter what was decided beforehand, concerning who wins and who loses, the player that comes in front of balanced configuration and its her turn to play, will always lose.

Let me say this one more time: It does not matter if it was decided that the player who takes the last object wins or that the player that takes the last object loses. Either way, if a player is in front of a balanced configuration and it is her turn to play, she will always lose the game.

Crazy, huh? Crazy, but correct. Let me explain why the above statements are correct.

I will study the two cases individually. The first case concerns the games where the player that takes the last object wins. The second case concerns the games where the player that takes the last object loses.

But before I do that, I have to stress the following points:

When a player is in front of a balanced configuration, and it is her turn to play, she will definitely create an unbalanced configuration. There is no other way, no matter how knowledgeable she is.

The mathematical reason for this is because she draws from only one line. Thus, something that was even among all lines, will stop being even.

When a player is in front of an unbalanced configuration, she has a choice. She can make the configuration balanced if she wants to, or she can produce a configuration that remains unbalanced.

Thus, when a player is in front of a balanced configuration, she is forced to make it unbalanced. But when a player is in front of an unbalanced configuration she has a choice of whether to balance it or to make it to remain unbalanced.

To be completely without choice when a player is in front of a balanced configuration is what creates the advantage for the other player. The other player can, with mathematical precision, literaly, win.

OK. Let me study the first case, the easy case, where the player that takes the last object wins.

This is the easy case.

So, the first player is front of a balanced configuration. From this, I know that the first player will definitely lose. Let us see why.

The first player will play and will make the configuration unbalanced. There is no other way. Even if she wants to, event is she does not want to, there is no other way for the first player but to leave the configuration unbalanced.

Then the second player will play and she will choose to balance the configuration.

Then the first player will play and she will make the configuration unbalanced again. Whether she wants to or not.

Then the second player will choose to balance the configuration, again.

And so on, until we reach the end of the game and the second player takes the last object, thus making the configuration balanced (zero objects is balanced, since zero is an even number).

So the first player is pushed around and guided unwillingly to the end of the game by the second player.

OK, the same thing happens with the second case. In the second case, too, the first player is pushed around and guided unwillingly to the end of the game by the second player.

But the analysis is way trickier in the second case.

OK. Let me study the second case, which is the case where the player that takes the last object loses.

So, the first player is front of a balanced configuration. From this, I know that the first player will definitely lose. Let us see why.

The first player will play and will make the configuration unbalanced. There is no other way.

Then the second player will play and she will choose to balance the configuration.

And so on, until we reach a very specific point in the game, where the second player will choose to reverse her strategy and leave the configuration unbalanced.

The reason why the second player will need to reverse her strategy should be obvious. If she doesn’t, we have the first case where she will end up taking the last object.

So, at a point in the game, the second player will need to reverse her strategy and leave the configuration unbalanced.

But this point in the game is very specific.

The second player must not choose to reverse her strategy at any point. No. In order to win, the second player must reverse her strategy, but at a very specific point and in a very specific way.

Which is this point and why?

Well, this point is the point where all remaining lines have only one object, with the exception of only one line which can have more than one object.

And why this is so? This is because only from this point onwards, the unbalanced configuration cannot be left unbalanced a second time by the first player.

If the second player would choose to leave the configuration unblanaced at a premature point, then the first player would balance it, bringing the second player at a disadvantage.

But if the second player leaves the configuration unbalanced at the specific point that I am talking about, from then on the configuration has to alternate between balanced and unbalanced.

Let me analyze this very subtle point: I described how from balanced we always go to unbalanced, but from unbalanced we go to balanced if we want to.

Well, yes, this is so, but from this specific point in the game onwards, from balanced we always go to unbalanced AND from unbalanced we always go to balanced.

So, from this specific point in the game onwards, if the second player reverses her strategy and leaves the configuration unbalanced, the first player cannot continue this trend but is instead forced to provide a balanced configuration.

Which the second player will make unbalanced and at last, the first player will make balanced by taking the last object, thus losing.

Do not be fooled that this very specific point (where the second player changes her strategy) is somewhere in the middle of the game. No. This specific point is at the very end of the game.

So, let us study this specific point.

As I said, this specific point is when all remaining lines (except one) contain only one object.

From this point onwards, from balanced we always go to unbalanced AND from unbalanced we always go to balanced.

And this is the point where the second player must reverse her strategy and leave the configuration unbalanced, for the first player to balance it (the first player has no other choice, whereas usually she would have a choice to let it remain unbalanced). And so on. From there we can only alternate between balanced and unbalanced, but not for long. We are already at the end of the game.

An example of this point is the following, The second player has to play to the following unbalanced configuration:

object

object

object

object object object

The second player will produce the following configuration, which remains unbalanced, and since it remains unbalanced, this is the change in her strategy that we were talking about:

object

object

object

From then on, it is the first player’s turn and the game is certainly in favor of the second player, as it was all along.

Another example of the point to change the strategy is the following:

object

object

object object object

Here, the second player will choose to produce the following configuration which remains unbalanced, and since it remains unbalanced, this is the change in her strategy that we were talking about:

object

object

object

Let me give you another example of this point where the second player has to change her strategy. We already said that this point is where all but one line have only one object.

So, let us suppose that the second player has in front of her the following configuration and is her turn to play:

object

object object

object object

We are near the end of the game. What must she do? She must change her strategy, yes? No, no yet. Remember, we said that the point where she needs to change her strategy is when only one line has more than one object. Here, two lines have more than one object.

So, she will need to continue with her initial strategy some more. So, she will still choose to balance the configuration:

object object

object object

Now, the first player will play, and unbalance the configuration.

Let us suppose that the first player produces the following unblanced configuration:

object

object object

Now this is the point where the second player will reverse her strategy and provide an unbalanced (instead of a balanced) configuration:

object

And the first player will play and take the last object, thus losing.

So, the second player had to change her strategy at the very end.

And this concludes my analysis.

The result of this analysis is that whenever a player has in front of her a balanced configuration, the other player has total control of the game from this point onwards and until the completion of the game.

So, this is a pointless game, just like tic-tac-toe.

In tic-tac-toe, if BOTH players know the ultimate playing strategy, the game will always end in a draw.

In NIM, if BOTH players know the ultimate winning strategy, the game will always end in favor not for the player that first comes in front of a balanced configuration to play, but in favor of the other player.

So, this game is pointless. But there is a way that this gane can be “saved” from being pointless. We can add chance to it, by using a dice, and one may, for example, take the number of objects denoted by the number which she rolls.

OK, so I have described how the game works and what the ultimate winning strategy is, but why does it work this way? Why do we need to use binary representations? (Equivalently we could have also used powers of two, and mentally consider the objects in groups of 1, 2, 4, 8, 16, 32, and so on, in each row, it is the same thing.) So, why do we need to transcend to the binary realm or to the powers of two to generate and use the ultimate winning strategy?

This is because we have 2 players. We have 2 players and they alternate playing. Player1 plays, then player2, then player1, then player2 and so on. One time we go from a balanced to an unbalanced configuration and the next time we may go from an unbalanced to a balanced configuration, if the current player wants to.

This means that we need to mind the binary representation’s 1’s as pairs. When a player plays, at least one of these 1’s is destroyed. The other player then has to aknowledge this fact and play accordingly.

And the ultimate winning strategy of the game is that since 2 players play, they need to consider these 1’s as pairs in each column for the configuration to be balanced.

]]>The purpose of the puzzle is to decode a number sequence in order to find the underlying word.

Let me explain.

First of all, this puzzle is meant to be case insensitive.

We begin with the most obvious correspondence: the 26 letters of the alphabet and the numbers from 01 to 26. So, the letter A corresponds to the number 01, the letter B corresponds to the number 02, and so on up until the letter Z which corresponds to the number 26.

Now we choose a word, whatever word we want. It would be best if it is a word with many letters. Why this is so, will become apparent later.

Then we substitute each letter of the original word with its corresponding number.

If we stop here, then it would be the easiest task in the world to decode the word from the number sequence.

But we do not.

What we do next is to change each number, advancing its value in a cyclical way. The change is as much as the value of the previous number in the sequence. And the first letter of the word, well, we change that according to the value of the last number. This is depicted in the following image.

What does advancing in a cyclical way means? It means that if we have the number 24, then advancing it by 01 will give us the number 25. Advancing it by 02 will give us the number 26. Advancing it by 03 will give us the number 01. Advancing it by 4 will give us the number 02. And so on. We go from 01 up to 26 and then we wrap back and continue from 01 onwards again.

Let me give you an example.

First of all, it will help if I write down the correspondence between letters and numbers.

01 A

02 B

03 C

04 D

05 E

06 F

07 G

08 H

09 I

10 J

11 K

12 L

13 M

14 N

15 O

16 P

17 Q

18 R

19 S

20 T

21 U

22 V

23 W

24 X

25 Y

26 Z

Suppose we want to code the word “JUSTICE”.

We begin with the letter sequence:

J U S T I C E

Then we create the initial number sequence:

10 21 19 20 09 03 05

This is easy to do. The letter J corresponds to 10, the letter U corresponds to 21, and so on.

Now we begin the difficult part. We will change the value of each number according to the number that precedes it. And for the first number, we will change its value according to the last number.

Let’s begin.

It is important to keep the initial number sequence intact and create a new one to be the result of our operations.

So let us begin by changing the second number.

The second number is 21. The number that precedes it is 10. So we will advance 21 by 10. Normally, this would give us 31. But we will not perform a regular addition. We will perform a circular addition where we wrap at 26. Thus 21 will become 05. So we will substitute 21 with 05.

The third number is 19. The number that precedes it is 21. So we will advance 19 by 21. Normally, this would give us 40. But since we are doing a circular advancement, 19 will become 14. So we will substitute 19 with 14.

The fourth number is 20. The number that precedes it is 19. So we will advance 20 by 19. Normally, this would give us 39. But since we are doing a circular advancement, 20 will become 13. So we will substitute 20 with 13.

The fifth number is 09. The number that precedes it is 20. So we will advance 09 by 20. Normally, this would give us 29. But since we are doing a circular advancement, 09 will become 03. So we will substitute 09 with 03.

The sixth number is 03. The number that precedes it is 09. So we will advance 03 by 09. Normally, this would give us 12. And since 12 is less than or equal to 26, we will accept this result as is. So we will substitute 03 with 12.

The seventh and final number is 05. The number that precedes it is 03. So we will advance 05 by 03. Normally, this would give us 08. And since 08 is less than or equal to 26, we will accept this result as is. So we will substitute 05 with 08.

Now let us change the first number. There is no number that precedes it, but we do everything circularly here, so we might as well continue in this path. We will use the value of the last number to advance the first number. So the first number is 10. And the last number is 05. So we will advance 10 by 05. Normally, this would give us 15. And since 15 is less than or equal to 26, we will accept this result as is. So we will substitute 10 with 15.

Please note that we could have done any of these operations in any order. We need to keep the initial number sequence intact. Then we can substitute each number with its advancement in any order. In the example above, I calculated the substitution of the first letter as the final operation, but I could have done it in the beginning or at any other time.

So let us see the final number sequence that we created:

15 05 14 13 03 12 08

Let us see the corresponding word:

O E N M C L H

So, if we give this word (that we calculated as our result) to someone, could they decode it? Could they derive the original word it came from, if the algorithm that we used is known to them?

It is important to keep the algorithm public and everyone should know the algorithm, meaning the procedure we used to derive the result word.

So, can someone derive the original word from the coded word? If someone knows our algorithm, and we give them the word OENMCLH, can they find out that the word it came from is JUSTICE?

Well, the result word does not resemble the original one, except for the fact that in this puzzle the initial word and the final word will always have the same number of letters. But even if the result word looks completely scrambled, we can reconstruct the original word it came from as follows:

We take the result word and we substitute the corresponding numbers.

So from

O E N M C L H

we come to

15 05 14 13 03 12 08

This is straightforward.

Next we can begin with any number and move in a cycle. So we might as well begin the first number.

The first number is 15. We know that we arrived at this number by augmenting the number that was there initially, using the last number in the sequence which is now 08. Yes, the last number is now 08, but we do not know what it was when we used it to augment the first number.

No number in this sequence is the original number that existed in the initial sequence. All numbers have been changed.

So what are we to do?

What we can do is suppose that the first number was originally 01. And see where this takes us.

Next we will suppose that the first number was originally 02. And see where this takes us.

And so on, until we check for all 26 possibilities for the first number.

Each time we suppose that the first number was something (from 01 to 26), we will come up with a number sequence. Then we will gather the 26 resulting number sequences and see which one corresponds to a valid word. The 25 other number sequences will be discarded. But it gets easier. Most, if not all, of the 25 number sequences, will not even be completed, since there will be an inconsistency when we wrap around the final to the first letter. So, most if not all of the 25 number sequences will be completely invalid. This is not because they will result in a meaningless bunch of letters, but because they will not result in a valid sequence that wraps validly around itself. This is because by supposing that the first letter was something, we will finally end up calculating the last letter, but when we derive this last letter, it should produce the assumption we made for the first letter. Usually, the only case that this will happen is when our original assumption for the first letter was correct.

Anyway, what we should do is assume each of the 26 possible numbers for the first number and calculate the original sequence that corresponds to this assumption. If we are able to finish the calculation and get a valid result, this is a strong hint that this result may correspond to the word we are looking for.

In the end, we will collect all the valid resulting sequences (theoretically their number will be from 1 to 16, but usually there will only be 1) and see to what words they correspond. The word we are looking for should be straightforward to find.

OK, let us see how we will proceed to find the original number sequence from the final number sequence

15 05 14 13 03 12 08

We will have to make 26 assumptions and then the calculations that correspond to each one of these assumptions.

So we will first assume that the original first number was 01.

So the corresponding calculated assumed original initial sequence is

01 …

If the original first number was 01 and now it is 15, this means that it was shifted by 14. So this means that the last letter was originally 14. We will keep that in mind. When we arrive at the end of our resulting original sequence and the last number is 14, this sequence will be a strong candidate for our final check. If not, this means that this sequence is definitely invalid, the assumption that the original first number was 01 is definitely invalid and we can safely discard this assumption and this resulting initial sequence.

By assuming that the first number was originally 01, we can deduce that the second letter was originally 04. This is because it is now 05 and its preceding letter was 01, so from 04 it was shifted to 05.

So the corresponding calculated assumed original initial sequence is

01 04 …

Now that we found the second number, we can find the third number. The third number is now 14, but since its preceding number was 04, then it means that it was originally 10 and it was shifted by 04 to become 14.

So the corresponding calculated assumed original initial sequence is

01 04 10 …

Now that we found the third number, we can find the fourth number. The fourth number is now 13, but since its preceding number was 10, then it means that it was originally 03 and it was shifted by 10 to become 13.

So the corresponding calculated assumed original initial sequence is

01 04 10 03 …

Now that we found the fourth number, we can find the fifth number. The fifth number is now 03, but since its preceding number was 03, then it means that it was originally 03 and it was shifted by 26 to become 03. In this circular addition, there is no zero as a number. We do not have any zeros. But if we add 26 to a number, we get the original number.

So the corresponding calculated assumed original initial sequence is

01 04 10 03 26 …

Now that we found the fifth number, we can find the sixth number. The sixth number is now 12, but since its preceding number was 26, then it means that it was originally 12 and it was shifted by 26 to become 12. I repeat from the previous paragraph: In this circular addition, there is no zero as a number. We do not have any zeros. But if we add 26 to a number, we get the original number.

So the corresponding calculated assumed original initial sequence is

01 04 10 03 26 12 …

Now that we found the sixth number, we can find the seventh number. The seventh number is now 08, but since its preceding number was 12, then it means that it was originally 22 and it was shifted by 12 to become 08. I repeat from the previous paragraph: In this circular addition, there is no zero as a number. We do not have any zeros. But if we add 26 to a number, we get the original number.

So the corresponding calculated assumed original initial sequence is

01 04 10 03 26 12 22

We finished, but… we found that the last number of the original sequence is 22. And we assumed that the first number was 01. Thus, in the final sequence it would have been shifted by 22 to become 23. But in the final sequence we have at hand, the first number is 15. Thus we arrived at an invalid resulting initial sequence. Thus, we can safely assume that 01 was not the first letter of the original initial sequence.

We can then assume that the first number of the original initial sequence was 02 and do all the calculations. And so on. Each time, we will keep the resulting initial sequence if it is valid, I,e, if the last number calculated validates the assumption of the first letter. And we will see which of the valid resulting initial sequences correspond to a legitimate word. But, usually, there will only be one valid resulting initial sequence.

So let us do the calculations for the correct assumption, to see how this goes.

The initial word was

J U S T I C E

and the initial number sequence was

10 21 19 20 09 03 05

But the solver does know that. All the solver knows is the final number sequence

15 05 14 13 03 12 08

Now let us suppose that the solver has tried all possible cases up from 01 up to 09 for the first number of the original sequence and now the solver is about to try the case for the first number of the original sequence being 10. We know that this is the correct assumption, but the solver does not. Let us see how this goes for the solver.

So the solver assumes that the original first number was 10.

So the corresponding calculated assumed original initial sequence is

10 …

The second number in the final sequence is 05. This means that the second number of the initial sequence was 21. This is because 10+21=05 in our circular addition.

So the corresponding calculated assumed original initial sequence is

10 21 …

The third number in the final sequence is 14. This means that the third number of the initial sequence was 19. This is because 21+19=14 in our circular addition.

So the corresponding calculated assumed original initial sequence is

10 21 19 …

The fourth number in the final sequence is 13. This means that the fourth number of the initial sequence was 20. This is because 19+20=13 in our circular addition.

So the corresponding calculated assumed original initial sequence is

10 21 19 20 …

The fifth number in the final sequence is 03. This means that the fifth number of the initial sequence was 09. This is because 20+09=03 in our circular addition.

So the corresponding calculated assumed original initial sequence is

10 21 19 20 09 …

The sixth number in the final sequence is 12. This means that the fifth number of the initial sequence was 03. This is because 09+03=12 in our circular addition.

So the corresponding calculated assumed original initial sequence is

10 21 19 20 09 03 …

The seventh number in the final sequence is 08. This means that the fifth number of the initial sequence was 05. This is because 03+05=08 in our circular addition.

So the corresponding calculated assumed original initial sequence is

10 21 19 20 09 03 05

And we finished. Now we must check the validity of the sequence. The last number produced is 05 and the first number is 10. Does 10+05 produce the first number of the final sequence? Yes, it does. 10+05=15, which is the first number of the final sequence.

Thus, the sequence we produced with the assumption that the first number is 10 is a strong valid candidate. And chances are slim to none that another valid sequence will emerge. But I suggest that the solver tries all 26 assumptions for the first letter of the initial sequence just in case.

So the corresponding calculated assumed original initial sequence is

10 21 19 20 09 03 05

And if we substitute the letters, we get the original initial word

J U S T I C E

Now this game can be played by choosing a word, encoding it using the circular algorithm I described and letting the solver deduce the way to decode the word. The solver should be able to deduce that she needs to try 26 possible assumptions for any letter position she chooses and move circularly calculating the resulting sequences. Of course, the solver can make the initial assumptions not for the first number but for any number in the sequence that she chooses. But there is no benefit or loss in what position she chooses to make the initial assumptions. But the solver has to deduce that she will have to make 26 assumptions and then perform the corresponding calculations for each assumption.

The word to be decoded should be long, in order to avoid letting the solver do the calculations by hand. This puzzle is meant to train aspiring cryptographers, so it would be best if it is designed such that it would persuade them to use a computer and create a program to do the calculations.

A student may create a program that decodes such a puzzle word and also a program that encodes (creates) such a puzzle word. And students can play against each other. One student can encode a word and the others could try and decode it.

Also, this puzzle can be made more difficult. For example, the symbols may be increased. As is, this puzzle deals with 26 symbols, the uppercase letters of the alphabet. Of course, a space, other symbols, the lowercase letters, numbers, etc. may be added. If we add, say, only a space, so the students can create an elementary sentence (just a space, no point, comma, etc) then the number symbols will be 27 and the addition will need to wrap at 27.

The difficulty of the puzzle can also be increased by circularly adding the previous, say, two letters to the current letter. And, in such a case, for the first letter, we would add the last and second before last letters. (Remember, we do things circularly here; not only the addition is circular, but also the way each letter is produced from the previous one – or more). So, if we choose to add the previous two letters to the current letter in order to shift it, we would have to make 26 *26 = 676 assumptions. So, to solve the puzzle we would need to calculate each assumption: each permutation of the first two letters, assuming we choose to work from the beginning of the number sequence. (Because, again, I am stressing that we can choose to begin to work from any number position in the sequence. And we can choose to proceed from left to right or from right to left.) Anyway, in such a case, we would assume that the first and second numbers in the initial sequence were 01 and 01. And we would find the third number, based on the third number of the final number sequence. And based on that, we would find the fourth number and so on until we calculated the whole sequence. And then we will try the assumption of the first number being 01 and the second number being 02. And so on, until we try all 676 assumptions.

So how may this puzzle appear? How can we pose the question to solve the puzzle? Here is an example:

**A word is encoded using the following algorithm. Each letter corresponds to a 2-digit number. The letter A corresponds to 01, the letter B to 02, all the way to the letter Z which corresponds to 26. By substituting each letter with its corresponding 2-digit number, an initial number sequence is produced.**

**Then, this initial number sequence is transformed as follows to produce a final number sequence. We take each 2-digit number and we add the previous (position-wise) 2-digit number. The resulting 2-digit number will be used in its place in the resulting number sequence. As far as the first 2-digit number is concerned, we will add the last 2-digit number to it to produce the first 2-digit number of the resulting number sequence. This concept is depicted in the accompanying image, where a six letter word is encoded.**

**In order for the resulting 2-digit number to always be from 01 to 26, we always perform a circular addition (instead of a normal one), where we wrap our results at 26. Thus, a few examples of this circular addition are 01+01=02, 01+25=26, 01+26=01, 10+22=06, 26+26=26, etc. So, 00 does not exist and adding with 26 equals the number that is added to 26.**

**Finally, in the resulting number sequence each 2-digit number is substituted with its corresponding letter. Given this resulting word, your task is to decode it, thus finding the original word.**

**So, given the following word, what was the original word that was used to produce it?**

**R X W W S R M Y Y C S**