The Haselbauer-Dickheiser Test can be found at http://matrix67.com/iqtest/.
In this blog post, I will study Question 11 from this test.
The question is about a game played in a hexagonal grid.
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 11.
Even though I have not really understood this question, I promise you, dear reader, that if you are patient enough to read this blog post, I am sure that you will find it useful nonetheless.
Question 11 is about a fictional game: Quess (I guess the name is coined so that it roughly reminds us of the game of chess). Quess is played on a specific hexagonal grid. In Quess, there may be many kinds of pieces, but we are informed about two of them: Warriors and Warp Gates. From now on, I will denote Warriors as red circles (disks) and WarpGates as blue circles (disks). Warriors may move up to seven spaces (hexagons) in a straight line, unless they encounter a WarpGate, in which case they warp to another WarpGate and continue their path in the same direction. WarpGates may move up to one space (hexagon) in any direction.
We are asked to find the minimum number of WarpGates that allow only four Warriors to attack the entire grid.
Already, a few questions come to mind:
When a Warrior encounters a WarpGate, the Warrior has to warp to another WarpGate. But can the Warrior choose which WarpGate it should warp to? Of course, if we use up to two WarpGates, this question does not apply. But what if we have more than two WarpGates? The rest of this blog post will assume that the Warrior is able to choose which WarpGate it will warp to.
We would like to use four Warriors to attack the whole grid. But why? Only one Warrior can attack the whole grid just as fine. A single Warrior can move only in a straight line. Once it stops, it can then move a gain to a different direction, thus visiting the whole grid. Do we want to have four Warriors, so that whatever space (hexagon) in the grid we choose, it can be immediately attacked by just moving one of the Warriors only once? The rest of this blog post will assume that.
If we assume that we want to have four Warriors able to reach any space (hexagon) in a single move, and we want to place the least number of WarpGates to achieve this, why does the question state that each WarpGate moves by only one space? Are we going to be moving WarpGates?
The rest of this blog post will assume that we will place four Warriors on the grid, trying to have them reach any one space (hexagon) in a single move. We will place the least amount of WarpGates in order to achieve that and we will not move any Warp Gate. Now, the question states how each WarpGate moves, but why it does so is a mystery to me. So, perhaps I have not understood the question. So, in the rest of this blog post, I might be solving the wrong problem.
For the following discussion,I found the following hexagon grid from Pratt & Larson.
I used the above grid to create the following Quess grid. You can use this Quess grid to try to solve the question.
The first thing I did when I created the Quess grid was to study it. For this, I drew straight lines, arriving at the following image. Please note that the following image contains no Warriors or WarpGates, just straight lines that can be drawn. This image helped me understand the way Warriors move on the grid.
From then on, I began to solve the problem. Keep in mind that I might be solving the wrong problem, since I have not understood the question.
Also, please note that in the following images, red circles denote Warriors, blue circles denote WarpGates, red lines denote paths a Warrior takes and blue lines denote paths a Warrior tales after it emerges from a WarpGate. Red paths (lines) always begin from a red circle. Blue paths (lines) always begin from a blue circle.
In addition please note that, in order to not clutter the images, I do not draw many parts of the red paths (especially if the hexagons where the paths lead have already been covered by another path) and I do not draw many blue paths. I do not draw only all lines and I do not draw them to their full extent. This way, it is easier to see the logic.
So, the first attempt I made is the following image, where there are four Warriors and they cover the whole grid except 6 hexagons. Now, if we place a WarpGate in each of these hexagons, then we have the whole grid covered with 6 WarpGates. But these 6 WarpGates are not used for warping, since they are not reachable by Warriors. These 6 WarpGate are only used to fill hexagons. Are we allowed to do that? That’s another question.
Moving on, the following attempt uses 2 WarpGates, but leaves 3 hexagons uncovered. May we fill them with more WarpGates?
The following attempt still uses 2 WarpGates, but leaves 2 hexagons uncovered. Better, but not adequate.
The following attempt uses 4 WarpGates. Here, the whole grid is covered and I present this as a solution.
Here is how the above grid is covered:
If the Warrior from above enters the middle left WarpGate, it will warp to the right WarpGate and continue downwards.
If the Warrior from above enters the middle right WarpGate, it will warp to the left WarpGate and continue downwards.
If the Warrior from below enters the middle left WarpGate, it will warp to the right WarpGate and continue upwards.
If the Warrior from below enters the middle right WarpGate, it will warp to the left WarpGate and continue upwards.
So, with 4 WarpGates, we have covered the whole grid. Can we do it with less?
I am not certain. I doubt that it can be done with 0 WarpGates. What about with 1 WarpGate? Well, once a Warrior enters a WarpGate, it has to emerge from another WarpGate and not the same. So, if there would be a single WarpGate, it would have to be unreachable to the Warriors and it will be used just to fill a hexagon. What about with 2 or 3 WarpGates? I cannot exclude this possibility. All I was able to do is to solve the problem with 4 WarpGates and the solution is in the image above.
Of course, as I have already mentioned, I might be solving the wrong problem.
Now, this is one of the many questions in the test that the test taker can find the answer without understanding or solving the question, thus gaming the system. The test taker can answer 0, then 1, then 2 and so on until the system rewards her.
I guess this is why the creators of the test say that the average person answers less than two questions correctly. They look at the answer submissions. But these submissions are made frantically from people who try to game the system to reveal the answers. So, a person would submit again and again, giving purposely wrong answers to all questions except the one she is trying to guess. This way, using the system as a oracle, she can deduce the answer to many questions that do not have a free form. And most of the questions do not. So, the system can be gamed easily. What the creators of the test witnessed when they reviewed the submission logs were not honest attempts by average people but coordinated attempts by people who were gaming the system and were successful at that, as well. Each attempt had as target to find the answer to only one question, thus all other answers were purposely incorrect. To the creators of the test, this seemed like the people that were submitting were trying but could only get one correct answer at most. The creators of the test got the wrong impression, because they thought that the submissions in their logs were honest ones. Little did they know that a large number of test takers succeeded in gaming and abusing their system.
At least, that’s my theory. When the system was live I knew nothing about it. I never used it. So, I can only speculate as to how it worked and how the submissions of the answers were made.
Anyhow, I do not like the fact that I tried to answer this problem (question 11), only to find that I had questions about it and no one to answer them. And I do not like the fact that the answers are not provided. Knowing but not telling: I find it to be a childish behavior.