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Mensa 3rd Graders At Recess

There are dozens of variations of this puzzle type, but I put my own spin and characteristics to it.

It is lunch time for the little ones at Mensa Mini's Private School. The children have come up with a unique, fun game that will involve them all.

There is a stack of index cards cut into quarters, one side blank - the other side has the name of a color spelled out. Each student chooses a card, face down, so they do not know what they chose. They then apply a piece of tape to the back and adhere to their forehead.

Outside they sit in a circle. Each student can see the others' card, but not their own.

How would these little geniuses go about figuring out their color, and leaving on the next tune played? How would these little Einstein's figure out what color is not theirs? Putting yourself in the place of one of the students, let's see your logic to get out as a winner!

The teacher explains the rules.

Rules

1. The teacher will play a tune by hitting a musical triangle at constant, regular intervals.

2. The moment a student knows the color written on his card, he will leave on the next triangle note. If anyone is wrong, they lose and go inside.

3. The teacher explains that the game is not impossible.

HINTS

Hint #1: Colors appear more than once. If they didn't the kid with the color would never know.

Hint #2: If a kid sees only one of a color, he can assume he is wearing the same color.

Hypothetical Game To Test With

Game

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closed as too broad by Peregrine Rook, Omega Krypton, Rupert Morrish, boboquack, Glorfindel May 23 at 5:38

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Can they simply ask each other "what is my color"? $\endgroup$ – Bewilderer May 22 at 23:16
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    $\begingroup$ It follows from Hint 1, I think @Bewilderer $\endgroup$ – El-Guest May 22 at 23:32
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    $\begingroup$ Welcome to Puzzling! Hint 1 is necessary to solve the puzzle, so it shouldn't be a hint, right? Also, is this not essentially just the same as the blue eyes problem (which we have many questions about)? $\endgroup$ – Deusovi May 22 at 23:53
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    $\begingroup$ @Deusovi Rule 3 technically implies Hint 1. $\endgroup$ – LeppyR64 May 23 at 3:34
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    $\begingroup$ Possible duplicate of In the 100 blue eyes problem - why is the oracle necessary? $\endgroup$ – boboquack May 23 at 5:05
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This is essentially the same as the Blue Eyes Problem (stated by Randall Munroe here, with solution here; there's also another PSE question that explains the solution in much more detail here).


There's one condition that makes this game work that's not stated in the question, however: all students must be perfect logicians. That is, they must make all possible deductions, and leave if and only if they are certain of their color. If this condition is not given, there is no way to win the game because you cannot gain information based on people's action or inaction.

For the actual solution, first we need an important piece of common knowledge:

Each color must appear at least twice. If not, the game would be impossible for the person with the unique color, since they would not have any way of determining their color.

Now, the "blue eyes" logic works as normal, with a minor change:

If you see any color that appears once, you can guess that you also have that color on round 1 (since otherwise, it would be unique). Therefore any color that appears 2 times will leave on round 1.

If you see any color exactly twice that doesn't leave on round 1, you know that you must have that color as well (or they would have left), so you leave on round 2. In other words, any color that appears 3 times will leave on round 2.

If you see any color exactly 3 times that doesn't leave on round 2, you know that you must have that color as well (or they would have left), so you leave on round 3. In other words, any color that appears 4 times will leave on round 3.

And this logic continues inductively: any color that appears n times will successfully leave on turn n-1.

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  • $\begingroup$ Can I comment on your post, sir? If I may, in my thought process in wording this, I sort of implied that 'clue' that each color much appear twice by stating that the puzzle was not impossible to solve. Because it seems quite obvious that if , let's say, when the time came to you and the second remaining player, you would know your color does not match because he would have left. Possibly that is a lot of information to assume the puzzle player to figure out? Not sure, sir. First time here. Thank you so much for responding so quick! Wow this site is fantastic so far. $\endgroup$ – HeyMikey May 23 at 2:18
  • $\begingroup$ Amazing, Deusovi. I am thoroughly impressed with your puzzle solution. Am I to expect this level of intelligence, articulation, and politeness throughout my puzzles? Seriously, sir, you nailed it. And wow, I didn't realize there were so many variations! This is on the back page of a text book in a slightly different situation. I sure will do my pre-posting research to make sure my puzzle is very unique. next time I try. $\endgroup$ – HeyMikey May 23 at 2:22
  • $\begingroup$ @HeyMikey There could be some other way that the game was not impossible (say, there was some sort of gimmick, like everyone getting the same color, or colors being assigned based on the students' names somehow). We know that's not the case because of how the problem is presented (as a pure logic problem), and so we can deduce that there are two of each color -- but the students need to know that as well for them to be able to win the game. $\endgroup$ – Deusovi May 23 at 4:35
  • $\begingroup$ (And no need to be so formal -- thank you for your compliments on my answer, and again, welcome to the site!) $\endgroup$ – Deusovi May 23 at 4:40

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