How Many Squares Are There On A Chessboard?
There are 64 squares. Together they create another square. After that there are 2x2 squares (a1-a2-b1-b2, for example), 3x3 squares (a1-c3), 4x4, 5x5, 6x6 and 7x7 squares.
You can see from this diagram that many different squares of various sizes exist. To go with the single squares, there are different squares of 2x2, 3x3, 4x4, and on up until you move to 8x8 (the board itself is a square too).
We have previously noted that there are 64 squares on the chessboard. We can cross-check this with a bit of immediate arithmetic. There are 8 rows, and each row includes 8 squares. So the total number of single squares is 8 x 8 = 64.
Analyzing the total number of larger squares is a little more complicated, but a quick diagram will make it much easier. There are three 2x2 squares marked on it.
If we explain the position of each 2x2 square by its top-left corner, then you can see that to stay on the chessboard, this crossed square must remain within the shaded blue area. You can see that every different position of the crossed square will point to a different 2x2 square.
The shady spot is one square smaller than the chess in both directions (seven squares); hence, the chessboard contains 7 × 7 = 49 unique 2x2 squares.
It comprises three 3x3 squares, and the total number of 3x3 squares can be calculated similarly to the 2x2 squares. Similarly, if we examine the top-left corner of each 3x3 square (represented by a cross), we can see that the cross must remain within the blue-shaded area for the 3x3 square to remain intact
Its square would extend beyond the board's edges if the cross were outside this location. The shaded region is currently six columns wide and six rows tall. Therefore, 6 × 6 = 36 different positions for the top-left cross and 36 possible 3x3 squares.
To determine the number of larger squares, we follow a similar procedure. As the squares, we are counting the increased size, i.e., 1x1, 2x2, 3x3, etc., the shaded area in which the upper left portion resides becomes one square smaller in each direction until we reach the 7x7 square pictured above.
There are now only four possible placements for 7x7 squares, as indicated by the crossed square in the upper-left corner of the blue-shaded area. We may use the rationale presented thus far to build a formula for determining the number of potential squares on any size square chessboard.
If n represents the number of squares on each side of the chessboard, then there are n x n = n2 individual squares on the board, just as 8 x 8 = 64 individual squares on a standard chessboard.
We have observed that the upper left corner of 2x2 squares must fit inside a smaller square than the original board. In total, there are (n - 1)2 2x2 squares.
Each time we add a unit to the side length of the squares, the blue-hued region into which their corners fit decreases by one unit in each direction. Consequently, (n - 2)2 3x3 squares and (n - 3)2 4x4 squares exist.
This process can be repeated until the final, largest square is the same size as the entire chessboard. Generally, the number of m x m squares on an n × n chessboard is always (n - m + 1).
Therefore, the total number of squares on an n × n chessboard will equal n2 + (n - 1)2 + 1. (n - 2). 2 plus, plus 22 plus 12, or the sum of all the square numbers from n2 to 12 inclusive.
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