2000 AMC 12 Problems/Problem 16

Problem

A checkerboard of rows and columns has a number written in each square, beginning in the upper left corner, so that the first row is numbered , the second row , and so on down the board. If the board is renumbered so that the left column, top to bottom, is , the second column and so on across the board, some squares have the same numbers in both numbering systems. Find the sum of the numbers in these squares (under either system).

Solution

Index the rows with i = 1, 2, 3, ..., 13 Index the columns with j = 1, 2, 3, ..., 17

For the first row number the cells 1, 2, 3, ..., 17 For the second, 18, 18, 19, ..., 34 and so on

So the number in row = i and column = j is f(i, j) = 17(i-1) + j = 17i + j - 17

Similarly, numbering the same cells columnwise we find the number in row = i and column = j is g(i, j) = i + 13j - 13

So we need to solve

f(i, j) = g(i, j) 17i + j - 17 = i + 13j - 13 16i = 4 + 12j 4i = 1 + 3j i = (1 + 3j)/4

We get (i, j) = (1, 1), f(i, j) = g(i, j) = 1 (i, j) = (4, 5), f(i, j) = g(i, j) = 56 (i, j) = (7, 9), f(i, j) = g(i, j) = 111 (i, j) = (10, 13), f(i, j) = g(i, j) = 166 (i, j) = (13, 17), f(i, j) = g(i, j) = 221 D. 555

See also

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