Problem:
It is possible to place positive integers into the twenty-one vacant squares of the square shown below so that the numbers in eaçh row and column form arithmetic sequences. Find the number that must occupy the vacant square marked by the asterisk .
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Solution:
Let and denote the numbers in two of the squares as shown in the first figure below, and compute the two neighboring entries in terms of them. Then the common difference in the third row is , while in the fourth row it is . Consequently,
Solving these equations simultaneously, we find that and . Therefore, the entries in the third and fourth row, and then in the fourth column may be computed to find that the number in the square marked by the asterisk (*) is . For completeness, the second figure shows the rest of the entries as well.
The problems on this page are the property of the MAA's American Mathematics Competitions