sorry if the question is too noob for this forum.. my 11 years old son was assigned to the task of solve a (kind of) magic square, that can be represented like the following system of equations:
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row $1: 6 + a + b = 11$
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row $2: c + d + 5 = 15$
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row $3: e + 4 + f = 19$
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col $\;\, 1: 6 + c + e = 15$
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col $\;\, 2: a + d + 4 = 16$
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col $\;\, 3: b + 5 + f = 14$
We solved it by writing a small program with a brute-force approach, but I wonder if there is some algebraic method, something more elegant that brute-force but simple enough to be explained to a kid (i. e. simplex matrix manipulations are too complex). Thanks for any suggestion!
Best Answer
So we have this magic square (the left with letters, the right with empty spaces):
Remaining numbers for filling the empty spaces are $1, 2, 3, 7, 8, 9$.
Let's begin with the first row. The sum of empty spaces in it must be $5 \ (=11-6)$. With the remaining numbers there are only two possibilities how to write $5$ as a sum of two numbers:
$$5 = 2 + 3 \\ 5 = 3 + 2 $$
The first one is impossible:
because in this case must be in the place of "?" number $10$.
So there is only the second possibility:
No problem for $11$-year-old child to finish the task now.