November 2, 2014
Section K
Luke Xu
Aren’t You a Square?
Problem Statement
You are asked to find the total amount of squares included in a chessboard. This includes squares that
are of sizes up to the whole chessboard. For example, when given a chessboard that has dimensions
of 5x5, the squares that would be inlcuded would be the 1x1, 2x2, 3x3, and 4x4 squares, as well as the
5x5 square. Also, how do you know that there are no squares left that were not counted?
Process
I was given a chessboard that had 8x8 as dimensions. I first started to count the squares manually. I
figured out that the 1x1 squares would be equal to the “area” of the chessboard. Because of this, I did
8*8 to get 64 1x1 squares. As I continued to count and mark off groups of squares that I had already
counted. I realized that there was a pattern. The number of 2x2 squares was 49, the number of 3x3
squares was 36, and the number of 4x4 squares was 25. I noticed that these numbers of squares were
“squares” themselves. The square root of 64 is 8, the square root of 49 is 7, the square root of 36 is 6,
and the square root of 25 is 5. From this, I would continue to count the squares manually, but I would
predict the amount of squares that each dimension would offer.
Solution
In order to help myself visualize the information gathered, I decided to construct a table.
listDimensions = {1, 2, 3, 4, 5, 6, 7, 8};
listSquares = {64, 49, 36, 25, 16, 9, 4, 1};
listA = Transpose[{listDimensions, listSquares}]
{{1, 64}, {2, 49}, {3, 36}, {4, 25}, {5, 16}, {6, 9}, {7, 4}, {8, 1}}
Page 1 of 4
Text[Grid[
Prepend[listA, {"Dimensions (X by X)", "# of Squares"}], Alignment → Center,
Dividers → {{2 → True, 2 → True}, {2 → True, 2 → True}}, Spacings → {1, 1}]]
Dimensions (X by X)
# of Squares
1
64
2
49
3
36
4
25
5
16
6
9
7
4
8
1
I then found the sum of the total number of squares that an 8x8 chessboard had.
Sum[x ^ 2, {x, 8}]
204
The total number of squares inside of an 8x8 chessboard is 204. I know that I am not missing any
squares because as I counted manually, I noticed that I was skipping an extra row and column. For
example, when I was counting the number of squares for the 2x2 case, I noticed that I counted only 7
rows of 2x2 squares and 7 columns of 2x2 squares. This is because as I neared the end of either the
row (I counted row by row), there would only be a 1x2 block leftover which was not enough to make
another 2x2 square. Similarly in the 3x3 case, I was constantly leftover with 2x3 blocks which led to
there being only 6 rows and 6 columns of the 3x3 blocks.
This does not mean that all of the 49 2x2 blocks had to fit into the 7x7 space, or the 3x3 blocks fitting
into the 6x6 space. What I mean by this is that the bottom-left unit of the block fits. Because I was
counting from left -> right(for rows), and bottom -> top (for columns), The only block that would never
fall out of the available space would be the bottom-left block. If I overflowed while counting rows, the
blocks that would overflow would be the right ones. If I overflowed while moving up, then the blocks that
would overflow would be the top ones.
Because of this, I know that I am not missing any possible squares because all of the other “squares”
that could have been counted would not be squares because they would not fit the definition of a
square.
Additionally, the bottom-left blocks are all 1x1 unit. Because they were given only a 7x7 space (in the
case of a 2x2 square), the total 1x1 unit bottom-left blocks would be 7*7 which is 49.
Generalizations
I can generalize this problem by making a formula for finding the total number of squares that an X by X
board would have. This formula would be:
November 2, 2014
Section K
Luke Xu
Sum[x ^ 2, {x, y}]
1
6
y (1 + y) (1 + 2 y)
In this formula, y would be the length of 1 side of the square. Because the shape is a square, we do not
need to have another value of the width because the width is equal to the length in squares.
For example, let there be a 10x10 chessboard that was given to us. In order to find the total number of
squares, the value “10” would need to substitute “y” in the formula above.
1
(10) (1 + (10)) (1 + 2 (10))
6
385
I have placed the number “10” in parenthesis for ease of reading.
The total number of squares in a 10x10 chessboard is 385.
Additionally, if the need of a table were to be needed, we can generalize that by having “n” equal the
different dimensions of the squares that we need. For example, n would equal “6” if the dimensions of
the square needed to be found were 6x6. Going along with the example, n will equal numbers from
1~10 because we need to find the squares with dimensions 1x1, 2x2, 3x3, 4x4, 5x5, 6x6, 7x7, 8x8, 9x9,
and 10x10.
n = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10};
listExampleSquares = n ^ 2
{1, 4, 9, 16, 25, 36, 49, 64, 81, 100}
ReverseList = Reverse[listExampleSquares]
{100, 81, 64, 49, 36, 25, 16, 9, 4, 1}
listExample = Transpose[{n, ReverseList}]
{{1, 100}, {2, 81}, {3, 64}, {4, 49}, {5, 36}, {6, 25}, {7, 16}, {8, 9}, {9, 4}, {10, 1}}
Page 3 of 4
Text[Grid[Prepend[listExample, {"Dimensions (X by X)", "# of Squares"}],
Alignment → Center,
Dividers → {{2 → True, 2 → True}, {2 → True, 2 → True}}, Spacings → {1, 1}]]
Dimensions (X by X)
# of Squares
1
100
2
81
3
64
4
49
5
36
6
25
7
16
8
9
9
4
10
1
Self-Assessment
From this chessboard problem, I believed that I have learned that there is a shortcut for calculating
problems similar to ones that ask for the total number of something. In this case, it was the number of
squares of a chessboard. An example of this would be like counting the number of triangles in a larger
triangle. I could apply the thought process that I had for this problem with the problem involving the
triangle. As far as diligence goes, I believe that I worked diligently on it. I spent roughly 3 hours on this
YGP, and I tried my best when explaining my thought process during the “Solution” stage. I did not
receive any assistance of any kind, unless googling commands for Mathematica is considered help.