Problem 3 · 1996 AJHSME
Easy
Arithmetic & Operations
pattern
The 64 whole numbers from 1 through 64 are written, one per square, on a checkerboard (an 8 by 8 array). The first 8 numbers go in order across the first row, the next 8 across the second row, and so on. After all 64 numbers are written, the sum of the numbers in the four corners will be
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Answer: A — 130.
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Hint 1 of 2
You don't need to fill in all 64 squares — only the four corners. Figure out which numbers land in the corners of the top row and the bottom row.
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Hint 2 of 2
Top row is 1–8 (corners 1 and 8); bottom row is 57–64 (corners 57 and 64). Look for a pairing that lets you add them without four separate additions.
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Approach: pair the diagonal corners
- Top row runs 1–8, so its corners are 1 and 8. The bottom row runs 57–64, so its corners are 57 and 64. Those four are all we need.
- Pair the diagonally opposite corners: 1 + 64 = 65 and 8 + 57 = 65. The total is just 2 × 65 = 130.
- Why this transfers: the smallest and largest of a balanced grid always pair to the same total — a 'sum from both ends' shortcut you'll reuse for arithmetic-series and symmetric-table problems.
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