How Many Squares


I have seen a lot of these going around on FaceBook.

It surprises me so many people cannot get the right answer, or often even get close. Here is how I break these down.

First start with the smallest squares. These are formed by the overlapping squares and we will call them 1 unit squares. There are 8 squares that are 1 unit by 1 unit (1 x 1 squares) or 4 each in 2 clusters. So our formula, looking at the square below is

  • (4 x 2) ==
  • 8

1by1

Next we got to 2 x 2. There are 4 squares and 4 rows, but we also have to take in account the 2 that overlap. So we have (4 x 4) + 2 = 18. Add in the original formula and we have

  • (4×2) + ((4×4) + 2) ==
  • 8 + (16 + 2) ==
  • 8 + 18 =
  • 26
  • So we now have 26 squares, as shown below.

2x2

There are no 3×3 squares

3x3

So the next size to look at is 4 x 4. To dissect this, you have to consider there are overlaps. This is what most people see, at first:

4x4_1

But there is an extra row:

4x4_2

And an extra column:

4x4_3

So the answer for the 4 x 4 squares is 3 overlapping rows by 3 overlapping columns (3 x3 = 9). So our answer now breaks down as

  • (4×2) + ((4×4) + 2) + (3×3) ==
  • 8 + (16 + 2) + 9 ==
  • 8 + 18 + 9
  • 35

There are no 5×5 squares

5x5

So we have to consider the 6 x 6 squares (those comprise of 3 of the 2 unit squares).

6x6

There is an overlap here, due to the size of the square:

6x6_2

So there are 2 squares on each row and 2 rows (see above and below):

6x6_3

This means there are 4 squares (2×2). The formula is now

  • (4×2) + ((4×4) + 2) + (3×3) + (2×2) ==
  • 8 + (16 + 2) + 9 + 4 ==
  • 8 + 18 + 9 + 4 ==
  • 39

If you go to the next size (4×4), there is only one.

8x8

  • (4×2) + ((4×4) + 2) + (3×3) + (2×2) + 1 ==
  • 8 + (16 + 2) + 9 + 4 + 1 ==
  • 8 + 18 + 9 + 4 + 1 ==
  • 40

So the answer is 40.

Peace and Grace,
Greg

Twitter: @gbworld

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