Power balance
A chess board has 64 squares. Pick numbers between 1 and 64 and place these numbers on each of the squares of the chess board such that the value in every square is the average of the values of its neighboring squares.
Prove that the only way to make such an arrangement is when all squares have equal numbers.
(http://www.cisl.columbia.edu/grads/tuku/brainteasers.html)
Prove that the only way to make such an arrangement is when all squares have equal numbers.
(http://www.cisl.columbia.edu/grads/tuku/brainteasers.html)
posted by littlecow at 9/17/2005 10:09:00 AM
3 Comments:
This one is trivial -- this basically boils down to a solution of the laplace equation ( (laplacian) F = 0 ). Supposed you write down some numbers randomly on the grid and then replace each number by the average of all its adjacent ones, eventually, at steady state, you would end up with each location having the average and that is when you would stop (can be proved by contradiction too)..
-Satish
I had not noticed the analogy to laplacian and central differences until you pointed it - its beautiful!
Danke..
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