Fibonacci

Fibonacci on the checkerboard


In the next diagram (checkerboard) only the white checkers are put on the board. We want to play the following game. A move in this game is a capture of a white checker by a white checker. You may only take a checker in the forward direction. The challenge is to bring a checker to the opposite site. In general: for which values of n is it possible to reach the opposite site of a square nxn checkerboard?
On the checkerboard there are drawn Fibonacci numbers. On the dark squares of the n-th row (from the bottom) you see the n-th Fibonacci number. Notice that at each capture starting from the n-th row, a checker from the n-th row and a checker of the n+1-th row disappear from the board and are replaced by a checker on the n+2-th row. In other words, the sum of all numbers beneath the white checkers never change. The sum of all numbers beneath the checkers is always 4*(1+1+2) = 16. The highest row contains the number 21 and so can never be reached.
If one fills the lowest row by zeros and the lowest row but one by ones, then the sum of all numbers beneath the checkers is 4*(0+1+1) = 8, and the highest row but one shows the number 8. This shows that if the highest row but one is reached, then you need all checkers that are not on the lowest row. The highest row but one can indeed be reached.
In general: The highest row can never be reached for a square nxn checkerboard.
For a relation between Fibonacci numbers and bridge see Jim Loy's site.

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