GRE® Counting Methods Practice

Counting all possible squares in a chess board

This GRE quant practice question is a permutation combination problem solving question. A classic counting methods question.

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

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

Video Explanation

Scroll down for explanatory answer text

Explanatory Answer

Use these hints to get the answer

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

Number of squares in a chess board

There are more squares in a chess board than the 64 1 x 1 squares.
The squares start from 1 x 1 all the way up to 8 x 8.

Let us count them and find a way to add all of them.

  1. 1 x 1 squares - 8 squares across the width and 8 squares along the length = 8 * 8 = 64
  2. 2 x 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 x 2) squares.
  3. 3 x 3 squares - 6 squares across the width and 6 along the length = 6 * 6 = 36 (3 x 3) squares.
  4. 4 x 4 squares - 5 squares across the width and 5 along the length = 5 * 5 = 25 (4 x 4) squares.
  5. 5 x 5 squares - 4 squares across the width and 4 along the length = 4 * 4 = 16 5 x 5) squares.
  6. 6 x 6 squares - 3 squares across the width and 3 along the length = 3 * 3 = 9 (6 x 6) squares.
  7. 7 x 7 squares - 2 squares across the width and 2 along the length = 2 * 2 = 4 (7 x 7) squares.
  8. 8 x 8 squares - 1 square across the width and 1 along the length = 1 * 1 = 1 (8 x 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{n(n+1)(2n+1)}{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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