# Staircase Sum

The digits 1-9 have been written in the squares so that each row and column sums to 13. What is the value of n?

## Problem

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The digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 are to be written in the squares so that every row and every column of three squares has a total of 13. Two numbers have already been entered. What is the value of n?

If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.

## Student Solutions

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Let $a, b, c, d, e, f$ be the numbers in the squares shown. Then the sum of the numbers in the four lines is $1 + 2 + 3 + ... + 9 + b + n + e$ since each of the numbers in the corner squares appears in exactly one row and one column. So $45 + b + n + e = 4 \times 13 = 52$, that is $b + n + e = 7$. Hence $b, n, e$ are $1, 2, 4$ in some order.

If $b = 2$ then $a = 2$; if $b = 4$ then $a = 0$. Both cases are impossible, so $b = 1$ and $a = 3$.

This means that $n = 2$ or $n = 4$. However, if $n = 2$ then $c = 10$, so $n = 4$ and $c = 8$.

*(The values of the other letters are e = 2, d = 7, f = 6.)*