How Many Squares Are There In A Chess Board?

This GRE quant practice question is a permutation combination - combinatorics - problem solving question. A classic counting methods question that tests the calculation of number of squares in a chess board.

Question 6: How many squares are there in a chess board?

1. 64
2. 65
3. 4096
4. 1296
5. 204

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Explanatory Answer | GRE Combinatorics Question 6

Hints to get the answer

Step 1: There are more squares than the 64 1 x 1 squares. Step 2: Find out the number of 2 x 2 squares and so on up to 8 x 8 squares. Step 3: Add the result.

Number of squares in a chess board

There are more squares in a chess board than the 64 1 × 1 squares. The squares start from 1 x 1 all the way up to 8 × 8. Let us count them and find a way to add all of them.

1. 1 × 1 squares - 8 squares across the width and 8 squares along the length = 8 × 8 = 64
2. 2 × 2 squares - with the size of the square increasing by 1 square the number of squares across the width will be down to 7 and the ones along the length will also be down to 7. So, there are 7 × 7 = 49 (2 × 2) squares.
3. 3 × 3 squares - 6 squares across the width and 6 along the length = 6 × 6 = 36 (3 × 3) squares.
4. 4 × 4 squares - 5 squares across the width and 5 along the length = 5 × 5 = 25 (4 × 4) squares.
5. 5 × 5 squares - 4 squares across the width and 4 along the length = 4 × 4 = 16 (5 × 5) squares.
6. 6 × 6 squares - 3 squares across the width and 3 along the length = 3 × 3 = 9 (6 × 6) squares.
7. 7 × 7 squares - 2 squares across the width and 2 along the length = 2 × 2 = 4 (7 × 7) squares.
8. 8 × 8 squares - 1 square across the width and 1 along the length = 1 × 1 = 1 (8 × 8) square.

Therefore, the total number of squares in a chess board = 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204 squares. If you figured that the number of squares is the summation of squares of natural numbers up to 8, you could have used the formula \\frac{\text{n(n+1)(2n+1)}}{\text{6}}), where n = 8.

Try this variant

How many rectangles are there in a chess board? Hint: A rectangle comprises two horizontal lines and two vertical lines. Count the number of pairs of horizontal and number of pairs of vertical lines possible in a chess board to get the answer.

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