| MadSci Network: Other |
There are 64 (8 x 8) squares on a chessboard, and the first square has one penny, or 20 pennies, so, the number of pennies on any square would be 2 n-1 (where n is the number of the square). So the sum of all 64 squares worth of pennies would be written as:
There is a mathematical trick to solving this sum, which avoids a lot of calculations. If we start adding the pennies from the first few squares, we can begin to see a pattern:
1 + 2 = 3 for 2 squares 3 + 4 = 7 for 3 squares 7 + 8 = 15 for 4 squares 15+16 = 31 for 5 squares 31+32 = 63 for 6 squaresThe pattern is that the sum of all previous squares is one less than the number of pennies in the next square, or the sum of the pennies on n number of squares is equal to one less than the number of pennies on the n+1 square, or the summation of a series of n doublings is equal to 2n - 1. So, for 64 doublings (chessboard squares), the sum (number of pennies) is 264 - 1, which is 18,446,744,073,709,551,615 pennies.
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