My attempt at Question 22 from the Haselbauer-Dickheiser Test

The Haselbauer-Dickheiser Test can be found at http://matrix67.com/iqtest/.

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:

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 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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Posted in Education

The ultimate winning strategy in the game of NIM

In the game of NIM, a few rows of objects are laid out. Each row can contain a number of objects, not necessarily the same number as the other rows.

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.

Posted in Science

The chained letters puzzle

I just invented a type of puzzle that I call “chained letters”.

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.

The chained letters puzzle

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.

The chained letters puzzle

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

Posted in Education

My attitude towards the flat Earth conspiracy theory

People who claim that the Earth is flat rather that spherical, and people that just are not sure whether Earth is flat or spherical, all they want is scientific proof in order to believe this or anything else for that matter.

And this is the mentality that science teaches us to have.

Richard Feynman said “Religion is a culture of faith. Science is a culture of doubt”. This means that in science, we welcome the doubters. Actually, we favor the doubters. Whenever something comes up from a person/scientist/whatever, the person invites universities, labs and everyone around the world to look into her findings and try to duplicate her experiment or validate her thinking to see if she was right. This is how science works.

So, if someone says to me that the Earth is flat or round, I should say, let me reproduce your claim. I should not say “all right”. I should say “fine, let me verify it.” This is how we teach Physics. We put students in the Lab. We do not say things to them and hope they remember them. We put them in the Lab. We give them microscopes, telescopes, timers, paper and blackboards to do calculations. All these in order for them to verify or dismiss theories.

So, if someone says to me the Earth is round and I take it for granted, then I am not a good scientist.

And if someone says to me the Earth is flat and I take it for granted, then I am not a good scientist.

Also, and this is very important, if someone says to me the Earth is flat and I laugh at her, then I am not a good scientist either. Good scientists do not mock others. Good scientists present their findings. Good scientists educate and inform others.

Mainstream scientists ask: “How can there be people who believe the Earth is flat after all the findings we present, like photo’s from outer space of the moon landing?”

Good question. I am not laughing at these mainstream scientists and I am not mocking them, even though I know the answer to their question.

The answer is that they have to be careful to understand what the flat-Earth-conspiracy-theorists are really saying.

What the flat-Earth-conspiracy-theorists are saying is: “Forget all the findings you presented. Forget them. I do not believe them. All these findings are conspiracy theories made exactly in order to fool us and to make us believe that the Earth is round.”

Mainstream scientists may ask: “Why would someone fabricate such a lie? Why would someone conspire to persuade others that the Earth is round?”

To the last question, the flat-Earth-conspiracy-theorists might provide very interesting and/or irrefutable answers, but our discussion is not about if and why they might be correct or incorrect in their suspicions.

Our discussion does not concern whether there is a conspiracy or not and why they would believe such a thing. Our discussion is not whether they are correct or incorrect that a conspiracy exists.

Our discussion has to do with what these people are saying. And these people are saying that the Earth is flat.

But mainstream scientists have to understand that the flat-Earth-conspiracy-theorists are basing their claim by disregarding all the facts that mainstream science presents as facts. And by doing this, they act as any real scientist would act.

So, I and anyone in the scientific community should welcome these people. Because they are not doing anything that science forbids us to do. Instead they are doing what science welcomes us to do: doubt.

So, these people doubt what we present as fact. And they say: “Forget about what they taught you. If you were to forget everything you were taught and start your reasoning from scratch, what would be your claims?”

So these people argue that if we observe and reason, we will come to the conclusion that the Earth is flat. But if we watch the news or get information from others, we will come to the conclusion that the Earth is round.

I know many of their arguments and observations and reasoning. And I know many of the arguments and observations and reasoning of mainstream science.

And I will tell you my opinion: the Earth is round, but it is extremely difficult, even for a Physicist, to prove it, when they are up against an informed and well educated flat-Earth-conspiracy-theory-supporter. (For example, one cannot say “here is photo from outer space”, because they will tell you that you did not personally take it, so it may as well be fabricated.)

Which makes these people worthy adversaries, people who deserve our respect and honor and whose claims are extremely difficult to falsify. But it can be done, flat-Earth-conspiracy-theory-supporters can be proven incorrect, and this is one of the reasons why we need to educate ourselves as much as possible in science.

In “The Art of War”, Sun Tzu writes about the mistake of underestimating your opponent.

I, for one, do not underestimate the flat-Earth-conspiracy-theory-supporters. Quite the opposite. But there are indeed some mainstream scientists that underestimate the flat-Earth-conspiracy-theory-supporters. I do not think highly of these mainstream scientists.

Posted in Science

Completely dynamic full screen overlay in JavaScript with div or canvas

In this blog post, I am going to show you how to create a full screen overlay.

Say you want to create a full screen popup when you click at a particular web page element, or at any other event you wish. And you wish for the popup to appear and to cover the browser’s full client area. And you also wish for the popup to disappear when you click anywhere on it.

Fear no more, your old pal Dimitrios (Actually, were we ever pals? Really, I mean, have I even met you?) is here to help you.

I will show you how to create a full screen overlay in JavaScript and I will give you the full html code for the whole html page. And I will show you how to do it by including support of the browser’s resize event. And I will show you how to do it using two methods.

The first method will be by dynamically creating a div. The second method will be by dynamically creating a canvas. The dynamic div or canvas will have some text and an image displayed on them, to enhancd the educational value of this post and to better showcase the value of these methods.

But which of these two methods should you use? Well, it depends. If you want to include html elements in the popup, then you should definitely use the dynamic div. This is because the hmtl canvas cannot display html elements. You have to draw everything yourself. You have to reinvent the wheel, so to speak. But if you do want to draw everything yourself, then you may use the dynamic canvas method.

Here are both methods, in all their glory, again, by your friend (Honestly, have we ever even just met?) Dimitrios:

Creation of a fully dynamic full screen overlay using an html div element

Creation of a fully dynamic full screen overlay using an html canvas element

Actually, when I began creating these methods, I would use a naive implementation where I did not account for the browser’s resize event. So I had all the code in one function. Thankfully, the implementations I gave above, account for the browser’s resize and scroll events.

I would also use an empty static element (div or canvas) with zero width. But the browsers would account for it ever so slighlty, even if it was empty and of zero length. This had as consequence that the underlying page would be slightly different when the popup disappeared than it was before its apperance. So, I decided to make everything completely dynamic. And, this way, I achieved perfect results.

Another point I would like to draw attention to is the use of  “return false;” in the click event in the undelying page. At first I did not include “return false;” and I could not get the popup to appear for more than what it seemed to me like a millisecond or so. Thus, without the “return false;” the popup would appear and disappear immediately. It would not stay at its place and it certainly would not wait for you to click in order for it to disappear. After a lot of thought, I realized that it was because I was directed to a new page (the same underlying page, that was actually reloaded) and to stop this from happening I had to include “return false;” to stop the redirection from happening. “return false;” lets us stay in the same page and it does not reload the page.

The last point I would like to make is for you to notice that I use canvas.width and canvas.height, whereas I use div.style.* and canvas.style.* for everything else, where “*” in this sentence means “any attribute”. But canvas.style.width and canvas.style.height are invalid ways to address these attributes (the width and the height of the canvas). At first, I made this mistake and, again, the canvas would disappear immediately. This time, it was not because the page was reloading, like it happened with the click event. This time, the script would encounter this error and would stop executing.

So, there you have it. I hope we are friends now!

Posted in Web design

About computer languages, IDEs, and human rights

I do not find tasteful or nice to have to use needy IDEs like Visual Studio. I like to use Notepad for programming.

Visual Studio is very demanding when it comes to resources like CPU, Memory, Hard Disk, Operating System version. You need to have the best and latest hardware and Windows version. I cannot keep up. I can never keep up. And even if I could, I feel that using such an IDE deprives me of functioning as a programmer. Whereas Notepad is the exact opposite.

I know it may sound crazy, but here is how I see it:

When AIDS first reared its ugly head, the news started advising everyone to use a condom during sex. All over the news and anyone you would talk to, would advise you to use a condom during sex. Of course, this is excellent advice that saves lifes.

At one point, CNN showed a discussion about this very matter. An expert was advising a group of men to use condoms during sex. At one point, one of the men said the following:

“We feel we are not functioning as men, if there is something in between.”

This is why I love CNN. They are the only ones that always show both sides of an argument. No one else had ever or ever again raised an argument against protection during sex.

Although I would urge everyone to use protection during sex, I will paraphrase the words of that man. And I will say:

I feel I am not functioning as a programmer if there is something (like an IDE) in between.

I want to have control over the code I write.

To me, my idea of programming is using Notepad.

To me, my idea of programming is to take a binary editor and construct the executable byte by byte.

Actually, I want to make choosing the programming language and the IDE one uses a human right.

Of course, a company would be correct to insist that all employees use a particular programming language and IDE or a particular set of them. But an individual should always have to right to resign because she feels she would like to use another programming language or IDE.

Again, the choice of programming language and way of programming should be a human right.

Posted in Development

Google doodle programming for kids solutions

Today (2017-12-04) Google published a doodle that has to do with programming for kids.

Although this is the way kids should learn how to program, the doodle is incredibly difficult and should address only seasoned programmers.

The way kids should learn how to program is exactly like that, though. You can create a floor with square tiles and stand in one. Then let the kids give you orders to guide you where they want you to go.

But Google should have never produced such a difficult puzzle meant for kids programming. This puzzle should have been addressed to adults.

Anyway, here are the questions:

Question 1:

Question 2:

Question 3:

Question 4:

Question 5:

Question 6:

And here are the answers:

Answer 1:

Answer 2:

Answer 3:

Answer 4:

Answer 5:

Answer 6:

Note:

By the way, the answer for question 5 can also be used to solve question 4.

Additional information:

Obviously, the goal is for the bunny to eat all the carrots.

The puzzle does not make it clear what considers as optimal solutions. From the image below, we can understand that optimal solutions are those which use the least amount of components. The number in each question denotes the minimum mumber of components needed to solve it.

All

Of course, since question number 4 can also be solved using the solution from question 5, one could argue that question 4 should have the number 6 (instead of 7) for denoting the components needed. But using the solution from question 5 to solve question 4 is far fetched, so I believe that denoting 7 as the compenent number for question 4 is better as it stands.

It is interesting to see what the optimal solution for question 6 is, if we consider as optimal solutions not those with the least number of components but those with the least number of bunny steps.
Then the answer to question 6 could be the following:
– – ) [ – ) – ( – ) – ]
where – means move one step, ) means turn clockwise, ( means turn counter-clockwise, and [ ] means repeat 4 times what is inside the square brackets. I depict this answer below, using two images, since it does not fit in one.

a.PNG

b.PNG

Final note:

Of course, the algorithms that the puzzle expects as optimal are unacceptable to me. If I would see someone creating these algorithms, I would tell them that they are far from safe.

You should be able to look at an algorithm and immediatelly deduce its validity, correctness, function, purpose, etc. You should be able to look at an algorithm and immediately understand what its results will be.

Programming is not about which programmer is more clever than the other. It is not an exercise in obfuscation. It is, or it should be, an exercise in clarity.

Posted in Development