One writes distinct positive integers into the cells of a table in such a way that, in each row and in each column, one number equals the sum of the other two numbers. Find the least possible total sum of the numbers written into the table.
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Solution
The sum of all numbers written in the cells is at least . As one number is the sum of others in every row, the sum of all numbers in every row is even. Thus the sum of all numbers in the table must be even, too. Hence the sum of all numbers in the table cannot be 45. It is possible that the sum is 46, as Fig. 6 shows.
Fig. 6
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