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How many Squares space There on a regular Chessboard?

So how countless squares space there top top a common chessboard? 64? Well, that course that is the correct answer if girlfriend are only looking in ~ the small squares lived in by the pieces during a video game of chess or draughts/checkers. However what about the bigger squares developed by group these small squares together? Look in ~ the diagram listed below to check out more.


Different size Squares ~ above a Chessboard

You deserve to see native this diagram the there are plenty of different squares of miscellaneous sizes. To go through the solitary squares over there are likewise squares the 2x2, 3x3, 4x4 and also so on up till you with 8x8 (the board itself is a square too).

Let's have a look at at how we deserve to count these squares, and also we'll also work the end a formula to be able to find the number of squares on a square chessboard of any type of size.

The number of 1x1 Squares

We have actually already detailed that there are 64 solitary squares on the chessboard. We can double-check this with a little of quick arithmetic. There room 8 rows and each row includes 8 squares, for this reason the total variety of individual squares is 8 x 8 = 64.

Counting the total number of larger squares is a little bit more complicated, but a fast diagram will certainly make the a lot of easier.


How countless 2x2 Squares space There?

Look at the diagram above. There room three 2x2 squares marked on it. If we define the place of each 2x2 square by its top-left corner (denoted by a overcome on the diagram), climate you deserve to see the to remain on the chessboard, this overcome square must remain within the shaded blue area. friend can also see the each various position of the overcome square will bring about a various 2x2 square.

The shaded area is one square smaller sized than the chessboard in both directions (7 squares) hence there room 7 x 7 = 49 different 2x2 squares top top the chessboard.


How many 3x3 Squares?

The diagram above contains 3 3x3 squares, and we have the right to calculate the total variety of 3x3 squares in a very similar method to the 2x2 squares. Again, if we look at the top-left edge of each 3x3 square (denoted by a cross) we can see that the cross must stay within the blue shaded area in order for its 3x3 square to remain fully on the board. If the overcome was exterior of this area, that is square would certainly overhang the edges of the chessboard.

The shaded area is currently 6 columns large by 6 rows tall, hence there room 6 x 6 = 36 areas where the top-left cross deserve to be positioned and so 36 feasible 3x3 squares.


What about the rest of the Squares?

To calculation the variety of larger squares, we proceed in the exact same way. Each time the squares we space counting obtain bigger, i.e. 1x1, 2x2, 3x3, etc., the shaded area the the height left component sits in becomes one square smaller sized in every direction until we reach the 7x7 square viewed in the picture above. Over there are currently only 4 positions the 7x7 squares deserve to sit, again denoted through the top-left overcome square sitting within the shaded blue area.

The Total number of Squares top top the Chessboard

Using what we have settled so far we deserve to now calculation the total variety of squares ~ above the chessboard.

Number that 1x1 squares = 8 x 8 = 64Number that 2x2 squares = 7 x 7 = 49Number the 3x3 squares = 6 x 6 = 36Number the 4x4 squares = 5 x 5 = 25Number that 5x5 squares = 4 x 4 = 16Number that 6x6 squares = 3 x 3 = 9Number of 7x7 squares = 2 x 2 = 4Number that 8x8 squares = 1 x 1 = 1

The total number of squares = 64 + 49 +36 + 25 + 16 + 9 + 4 + 1 = 204

What about Larger Chessboards?

We can take the reasoning that we have actually used so far and also expand upon the to develop a formula for functioning out the variety of squares possible on any kind of size of square chessboard.

If us let n represent the size of each side the the chessboard in squares climate it follows that there are n x n = n2 individual squares on the board, similar to there are 8 x 8 = 64 individual squares top top a typical chessboard.

For 2x2 squares, we have actually seen the the height left edge of these must fit right into a square the is one smaller than the original board, thus there are (n - 1)2 2x2 squares in total.

Each time we include one come the side length of the squares, the blue shaded area the their corners fit into shrinks by one in each direction. Therefore there are:

(n - 2)2 3x3 squares(n - 3)2 4x4 squares

And therefore on, till you obtain to the final huge square the exact same size together the totality board.

In general, you have the right to quite conveniently see that for an n x n chessboard the variety of m x m squares will always be (n - m + 1).

So because that an n x n chessboard, the total variety of squares of any kind of size will certainly equal n2 + (n - 1)2 + (n - 2)2 + ... + 22 + 12 or, in various other words, the sum of every the square numbers from n2 down to 12.

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Example: A 10 x 10 chessboard would have a complete of 100 + 81 + 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 385 squares.

Something come Think About

What around if you had a rectangle-shaped chessboard v sides of various lengths. How can you expand our thinking so much to come up v a means of calculating the total variety of squares on an n x m chessboard?