HMMT 二月 2021 · COMB 赛 · 第 1 题

HMMT February 2021 — COMB Round — Problem 1

Leo the fox has a 5 by 5 checkerboard grid with alternating red and black squares. He fills in the grid with the numbers 1 , 2 , 3 , . . . , 25 such that any two consecutive numbers are in adjacent squares (sharing a side) and each number is used exactly once. He then computes the sum of the numbers in the 13 squares that are the same color as the center square. Compute the maximum possible sum Leo can obtain.

解析

Leo the fox has a 5 by 5 checkerboard grid with alternating red and black squares. He fills in the grid with the numbers 1 , 2 , 3 , . . . , 25 such that any two consecutive numbers are in adjacent squares (sharing a side) and each number is used exactly once. He then computes the sum of the numbers in the 13 squares that are the same color as the center square. Compute the maximum possible sum Leo can obtain. Proposed by: Milan Haiman Answer: 169 Solution: Since consecutive numbers are in adjacent squares and the grid squares alternate in color, consecutive numbers must be in squares of opposite colors. Then the odd numbers 1 , 3 , 5 , . . . , 25 all share the same color while the even numbers 2 , 4 , . . . , 24 all share the opposite color. Since we have 13 odd numbers and 12 even numbers, the odd numbers must correspond to the color in the center square, so Leo’s sum is always 1 + 3 + 5 + · · · + 25 = 169.