Problem:
A checkerboard of 13 rows and 17 columns has a number written in each square, beginning in the upper left corner, so that the first row is numbered , 17 , 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 square have the same numbers in both numbering systems. Find the sum of the numbers in these squares (under either system).
Answer Choices:
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Solution:
Suppose each square is identified by an ordered pair , where is the row and is the column in which it lies. In the original system, each square has the number assigned; in the renumbered system, it has the number assigned to it. Equating the two expressions yields , whose acceptable solutions are , , and . These squares are numbered and 221 , respectively, and the sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions