# Number Properties Puzzle -Critical Thinking 1

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## Challenge:

How many squares are there in a chess board?

64? 65? No! Consider squares of 2 × 2, 3 × 3, etc.

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The correct answer is 204.

This question implies “squares of any size”. You can find squares of 1 by 1, 2 by 2, 3 by 3 and so on. A chessboard is a square of 8 by 8 and hence there are 64 (8 x 8 = 64) 1 by 1 square on a chessboard.

The next size square is a 2 by 2 unit square. First, find all 2 by 2 squares in the first two rows. You can find seven of them. Then, consider rows 2 and 3. Again, you can find seven. Consider rows 3-4, 4-5, 5-6, 6-7, and 7-8. You’ll find seven 2 by 2 squares for each pair. Therefore, there are 7 × 7 = 49 squares of 2 by 2.

We can use the same pattern for the squares of 3×3. You can find six squares of 3×3 in the first 3 rows. Then, consider rows of 2-3-4, 3-4-5, and so on. You’ll find 6 × 6 = 36 squares of 3 by 3.

Continue the pattern for other size of square.

64 squares of 1 × 1

49 squares of 2 × 236 squares of 3 × 3

25 squares of 4 × 4

16 squares of 5 × 5

9 squares of 6 × 6

4 squares of 7 × 7

1 square of 8 × 8

Total number of squares:

64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204

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