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Problem 187

Combinatorics Difficulty 1.5 Multiple choice CEMC Gauss (Grade 7) · Canada · 2015

In the $3 \times 3$ table shown, the numbers 1, 2 and 3 are placed so that each number occurs only once in each row and only once in each column.

The value of $X +Y$ is

Pick one

Official solution

Consider the value of $U$ in the diagram shown.

[[IMAGE0]]

Since a 3 already occurs in the second row, then $U$ cannot equal 3 (each of the numbers $1,2,3,$ occur only once in each row).

Since a 1 already occurs in the third column, then $U$ cannot equal 1 (each of the numbers $1,2,3,$ occur only once in each column).

Since $U$ cannot equal 3 or 1, then $U=2$ .

Therefore, a 2 and a 3 already occur in the second row and so $X=1$ . At this point, a 2 and a 1 already occur in the third column and so $Y=3$ .
The value of $X+Y=1+3=4$ .

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.