2021/22 AHSMC Part 1 Problem 3

Problem 3 of 16

The $25$ little squares of a $5$ by $5$ grid are filled with $1$ 's, $2$ 's and $3$ 's, with each little square getting exactly one number, and so that no two squares touching each other either horizontally or vertically get the same number.

Figure for this problem

The numbers in the little squares around the outside of the grid are as shown. What is the number in the centre square?

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Solution

Let $a_{ij}$ be the number which fills the row $i$ and column $j$ , where $i, j \in \{1,2,3,4,5\}$ . We first remark that $a_{24} = 2$ and $a_{44} = 3$ . Now, one by one, $a_{23} = 1$ , $a_{22} = 3$ , $a_{32} = 2$ , $a_{42} = 1$ , $a_{43} = 2$ , and $a_{34} = 1$ . Hence $a_{33} = 3$ . The answer is (C).

University of Alberta

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