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
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.
Alternatively, it is possible to see the patter as a complete
square with corners and a central square removed. So for the n
th pattern, we have a complete
$(n+4)(n+4)$ square with the four corners and a central $n\times n$
square removed. Hence the number of squares is $(n+4)^2 - n^2 -4 =
This problem is taken from the UKMT Mathematical Challenges.