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This one is my favorite, too—it really highlights what the job of a mathematician is. The board is the board, and the dominos either fit or they don’t, and it’s not clear why. But once someone adds the checkerboard shading—not changing the problem at all, but just adding a new way to look at it—suddenly the solution falls out, clear and obviously true.
I am curious. Are all boards that have equal white and black tiles counts solvable?
This one is my favorite, too—it really highlights what the job of a mathematician is. The board is the board, and the dominos either fit or they don’t, and it’s not clear why. But once someone adds the checkerboard shading—not changing the problem at all, but just adding a new way to look at it—suddenly the solution falls out, clear and obviously true.
I am curious. Are all boards that have equal white and black tiles counts solvable?
Bunch of problems like this are in the book "Problem-Solving Strategies" by Arthur Engel [1], I believe even including this particular one. Fun book. [1] https://www.google.com/books/edition/Problem_Solving_Strateg...
This problem reminds me of another problem. There is a round table and two players, A and B. The players take turns placing a coin on the table in any location they desire, but coins may not overlap. The first person who is unable to place a coin loses. What is the winning first move? Answer: the winning move is for player A to place the first coin in the center of the table. After that, no matter which location play…
And if they don't, then what?
This problem reminds me of another problem. There is a round table and two players, A and B. The players take turns placing a coin on the table in any location they desire, but coins may not overlap. The first person who is unable to place a coin loses. What is the winning first move? Answer: the winning move is for player A to place the first coin in the center of the table. After that, no matter which location play…
This problem reminds me of another problem. There is a round table and two players, A and B. The players take turns placing a coin on the table in any location they desire, but coins may not overlap. The first person who is unable to place a coin loses. What is the winning first move? Answer: the winning move is for player A to place the first coin in the center of the table. After that, no matter which location play…
They didn't consider it a valid solution :|
This problem reminds me of another problem. There is a round table and two players, A and B. The players take turns placing a coin on the table in any location they desire, but coins may not overlap. The first person who is unable to place a coin loses. What is the winning first move? Answer: the winning move is for player A to place the first coin in the center of the table. After that, no matter which location play…
I solved this in a hedge fund interview with "Consider the limiting case of a coin the same size as the table... You can only place it at the centre, and you win... Now make the coin smaller. How does the strategy change?... It can't, because of symmetry." They didn't consider it a valid solution :|
https://jackmorris.xyz/2015/the-simple-proof-of-the-tetris-l...
It goes like this (stolen from a website - there are many variations on this):
Problem: Two trains are on the same line, 60 miles apart, heading towards each other, each traveling at 30 mph. A fly that can travel at 60 mph leaves one engine flying towards the other. Upon reaching the other engine, it instantaneously turns around, and heads back to the other engine. This is repeated until the two trains crash and the fly is annihilated at the same time.
Question: How far does the fly travel before it is "splatted"?