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?
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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.)