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Algebra Difficulty 2.5 Junior Find the answer

The odd numbers from 5 to 21 are used to build a 3 by 3 magic square. If 5, 9 and 17 are placed as shown, what is the value of xx?

A number or a short expression. Spacing and $ signs are ignored.

Solution

The sum of the odd numbers from 5 to 21 is 5+7+9+11+13+15+17+19+21=1175+7+9+11+13+15+17+19+21=117. Therefore, the sum of the numbers in any row is one-third of this total, or 39. This means as well that the sum of the numbers in any column or diagonal is also 39. Since the numbers in the middle row add to 39, then the number in the centre square is 39−9−17=1339-9-17=13. Since the numbers in the middle column add to 39, then the number in the middle square in the bottom row is 39−5−13=2139-5-13=21. Since the numbers in the bottom row add to 39, then the number in the bottom right square is 39−21−x=18−x39-21-x=18-x. Since the numbers in the bottom left to top right diagonal add to 39, then the number in the top right square is 39−13−x=26−x39-13-x=26-x. Since the numbers in the rightmost column add to 39, then (26−x)+17+(18−x)=39(26-x)+17+(18-x)=39 or 61−2x=3961-2x=39 or 2x=222x=22, and so x=11x=11.

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