A large square is ruled into a $4\times 4$ arrangement of 16 identical small squares. How many squares of any size at all can be traced in the figure?

Count by size. Squares 1 cell wide: $4\times 4=16$. Squares 2 cells wide: $3\times 3=9$. Squares 3 cells wide: $2\times 2=4$. Squares 4 cells wide: 1. Total 16 + 9 + 4 + 1 = 30. The traps: 16 counts only the smallest squares, and 17 counts the smallest ones plus the outer square while missing every in-between size.