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One way to proceed is to regard the pattern as four arms, each two squares wide, with four corner pieces of three squares each. So for the $n^\text{th}$ pattern, we have $4 \times 2 \times n + 4 \times 3 = 8n +12$. For $n=10$, we need $8 \times 10 +12$, i.e. $92$ squares.

This problem is taken from the UKMT Mathematical Challenges.
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