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Warmup question: How many squares are there on a chessboard?

Edited:

Grand prize: How many rectangles are there on a chessboard?

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How many squares are there on a chessboard?

I like this one for it makes my head dizzy when i try to visualize it but the answer is ...

204

Say your chess board has all 1 unit squares. Now if you count the edge for all 1 unit square there would be 8 on each side => 64 squares in total.

Now count for the 2 unit squares. 7 on each side => 49 in total.

3 unit square. 6 on each side => 36 in total. and so on.

So Total Number = 1+4+9+16+25+36+49+64 = 204!

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square of size 8: 1 (1X1)

square of size 7: 4 (2X2)

square of size 6: 9 (3X3)

square of size 5: 16 (4X4)

square of size 4: 25 (5X5)

square of size 3: 36 (6X6)

square of size 2: 49 (7X7)

square of size 1: 64 (8X8)

in total : 204

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squares:

1 * 8x8

4 * 7x7

9 * 6x6

etc

x^2 * 9-x by 9-x

written out fully:

1 * 8x8

4 * 7x7

9 * 6x6

16 * 5x5

25 * 4x4

36 * 3x3

49 * 2x2

64 * 1x1

1+4+9=14+16=30+25=55+36=91+49=140+64=204

204 squares on a chessboard

as for the rectangles, this includes squares, right?

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1x1: 64

1x2 and 2x1: 112

1x3 and 3x1: 96

1x4 and 4x1: 80

1x5 and 5x1: 64

1x6 and 6x1: 48

1x7 and 7x1: 32

1x8 and 8x1: 16

2x2: 49

2x3 and 3x2: 84

2x4 and 4x2: 70

2x5 and 5x2: 56

2x6 and 6x2: 42

2x7 and 7x2: 28

2x8 and 8x2: 14

3x3: 36

3x4 and 4x3: 60

3x5 and 5x3: 48

3x6 and 6x3: 36

3x7 and 7x3: 24

3x8 and 8x3: 12

4x4: 25

4x5 and 5x4: 40

4x6 and 6x4: 30

4x7 and 7x4: 20

4x8 and 8x4: 10

5x5: 16

5x6 and 6x5: 24

5x7 and 7x5: 16

5x8 and 8x5: 8

6x6: 9

6x7 and 7x6: 12

6x8 and 8x6: 6

7x7: 4

7x8 and 8x7: 4

8x8: 1

Grand Total: 1296 :D

Am I right? Did I make any mistakes or miss anything?

Note: I figured this out by myself, so please do not take any credit.

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after a few seconds of thinking I came up with the following formula:

w = width of the rectangle

h = height of the rectangle

wa = width of the bigger rectangle

ha = height of the bigger rectangle

the formula for the amount of rectangles (squares included) of w width and h height is:

(wa+1-w)(ha+1-h)

In this case wa and ha are both 8, cuz the chessboard is 8x8, so this simplifies the equation to:

(9-w)(9-h)

Order matters, so all numbers 1-8 will be used for both w and h, which means adding together the 64 multiplication problems.

(9-1)(9-1) + (9-1)(9-2) + (9-1)(9-3) + ...etc

fortunately I made a program that did that for me lol

I got 1296

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... 1296 Rectangles

Rectangles of Width 8 : Num of Rectangles

X1 = 1*1=1

X2 = 1*2=2

X3 = 1*3=3

X4 = 1*4=4

X5 = 1*5=5

X6 = 1*6=6

X7 = 1*7=7

X8 = 1*8=8

...... In total 36 rectangles of width 8 units and varying height

Rectangles of Width 7 : Num of Rectangles

X1 = 2*1=2

X2 = 2*2=4

X3 = 2*3=6

X4 = 2*4=8

X5 = 2*5=10

X6 = 2*6=12

X7 = 2*7=14

X8 = 2*8=16

...... In total 72 rectangles of width 7 units and varying height

Rectangles of Width 6 : Num of Rectangles

X1 = 3*1=3

X2 = 3*2=6

X3 = 3*3=9

X4 = 3*4=12

X5 = 3*5=15

X6 = 3*6=18

X7 = 3*7=21

X8 = 3*8=24

...... In total 108 rectangles of width 6 units and varying height

....

In total 144 rectangles of width 5 units

In total 180 rectangles of width 4 units

In total 216 rectangles of width 3 units

In total 252 rectangles of width 2 units

In total 288 rectangles of width 1 units

Therefore the number of rectangles is 36+72+108+144+180+216+252+288 = 1296

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