A subset $S$ of the set of integers 0 - 99 is said to have property $A$ if it is impossible to fill a crossword-puzzle with 2 rows and 2 columns with numbers in $S$ (0 is written as 00, 1 as 01, and so on). Determine the maximal number of elements in the set $S$ with the property $A.$
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The condition is equivalent to:
No digit may appear both as the first digit of some number and as the last digit of some number.
Let $x$ be the number of digits which occur only as the first digit. Let $y$ be the number which occur only as the last digit. The number of possibilities is then $xy$.
Now we have $x+y=10$, so $xy \leq \frac{1}{4}(x+y)^2 = 25$. Equality occurs when $x=y=5$.