The website of my university posted a riddle that goes something like this:
There are three men named 1,2 and 3 and each one has two colored dots on his forehead. Possible colors are black and red. No color is used more than four times. The men can see the colors on the other mens head, but not the ones on their own. A game-master asks the men in the order 1,2,3,1,2,3,… whether they know the colors of their dots. If one says “no”, he proceeds with the next man. If one says “yes”, he has to state his guess. If he is correct, all men win the game, if he is wrong then all men lose. The men were not given any time to make up a strategy.
As it turnes out the men answer “no”, “no”, “no”, “no”, “yes”. What is the color on the head of man 2?
I was going to solve this riddle systematically by trying to reason what color combinations make sense and what information each man can deduce from the others saying “no”.
However, I came to a much more direct solution, that completely ignores most of the riddle and only works by the assumption that the riddle is solvable.
Solution: Man 2 must have a red and a black dot on his forehead.
Reason: I (from the meta perspective) only have the information about the answer-pattern and no actual colors. Since the game is symmetric w.r.t. the colors red and black, the game would have the same answer-pattern if we switch the colors. This means, any reasoning that explains that man 2 has e.g. two red dots, will also work to reason that he has two black dots. If any of these two would be the solution, I would not be able to conclusively find it. Hence the solution must be “one red and one black dot”. Note that I cannot answer how man 2 came to his conclusion, but this was not part of the question.
For me, this kind of solving the riddle was very interesting and I wondered if there are more such riddles out there which (intentionally or unintentionally) can be solved by such meta-assumptions.
Question: Are there more such riddles which have a surprisingly easy solution by meta-assumptions like e.g. “the riddle has a solution” or “the riddle is solvable with reasonable effort” etc.
Because it was asked in the comments, and maybe needs a clarification, here is what I am looking for in other words:
I ask for riddles or math-problems
- which are formulated in a way that suggests some specific meta-assumption (e.g. the riddle must have a solution which I as a reader can find).
- the meta-information is not obviously presented to the reader as something that should be used to solve the riddle (preferably might not even be needed).
- the meta-information turns out to be unexpectedly helpful.
I am not specifically asking about the assumptions of “existence of a unique solution”, but also other meta-assumptions, like e.g. the context of the question, the number of answers (if it is multiple-choice), the expected time-frame of solution, maybe even the color of the paper the question was asked on, etc.
Some type of examples which came up a lot (and which I liked) are riddles/problems where the meta-information was the absence of some information. The absence implied that the solution will probably not depend on this missing information and hence we can choose a very simple instance of the problem.
One other example that came to my mind (but it does not fit so perfectly) is the graph-problem presented in this video of 3Blue1Brown. It can (and must) be solved by the meta-information that it is presented on a cup and not on a piece of paper. Unfortunatly the riddle can only be solved by recognizing that it is on a cup. I would prefer that the meta-way is not the intended solution.
Having solved my share of logic puzzles, I can report that logic puzzles can be solved using the meta-assumption that the logic puzzle is solvable surprisingly often (in fact, I had the occasion of ‘solving’ the very puzzle you mention just a few months ago in exactly the way you did). I use ‘solving’ since of course you are not really solving the riddle at all, but merely showing that ‘if the riddle has a solution, then it has to be …’. Here is another example:
In the Sudoku-solving (and other grid-filling puzzles) community the meta-assumption that a Sudoku puzzle has a unique solution is called the Uniqueness assumption and can be used to great effect. For example, suppose you know that you have a row where there are two open cells left (say in columns 2 and 3) which must contain the numbers 1 and 4. Suppose you have another row with three open cells left (say columns 2, 3, and 5), which must contain the numbers 1, 4, and 7. Then using the Uniqueness assumption we can immediately say that the cell in that second row in column 5 does not contain a 7, since if it did, you could place the remaining 1‘s and 4‘s in two different ways, and clearly this has no bearing on the solvability of the rest of the puzzle.