As promised, here is the 1st of 3 Checkerboard puzzles by one of our longstanding regulars, Bobo The Bear. This first problem has been around for a long time, and might be recognized by several of you. The 2nd and 3rd, however, are originals : )
How many squares, of all sizes, are there on a standard 8×8 checkerboard?
As always, answers can be entered in the comment section below. Answers to be revealed next week, thanks
Size 8: 1×1
Size 7: 2×2
Size n: n^2
Total = Sum for i=1 to 8 (i^2) = 204
1, 8×8 square
4, 7×7 squares
9, 6×6 squares
16, 5×5 squares
25, 4×4 squares
36, 3×3 squares
49, 2×2 squares
64, 1×1 squares
i.e. 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204 squares
If my math is right here, we have 64 1×1 squares, 49 2×2, 36 3×3, 25 4×4, 16 5×5, 9 6×6, 4 7×7, and 1 8×8.
64+49+36+25+16+9+4+1=204
Cool puzzle, I did it from an example:
In a 2X2 checkboard= 2^2 +1 = 5 squares of all sizes
In a 3X3 checkboard= 5(squares of all sizes in a 2×2 square) + 3^2= 14
and so on…..
so in a 8X8 checkboard you have: 1 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2
—> 221 (squares of all sizes)
204 (64 + 49 + 36 … + 1)
Squares on a checkerboard:
1) 8*8=64
2) 7*7=49
3) 6*6=36
4) 5*5=25
5) 4*4=16
6) 3*3=9
7) 2*2=4
8) 1*1=1
TOTAL = 204
Doing an engineering analysis I came up with 204 squares. I have a spreadsheet with graphical representation of all squares counted. I don’t have a website, so I cannot give you a link to it.
Here’s the text part:
64 1×1
16 2×2 (top and left)
12 2×2 (top and right 1)
12 2×2 (down 1 and left)
9 2×2 (down 1 and right 1)
4 3×3 (top and left)
4 3×3 (top and right 1)
4 3×3 (top and right 2)
4 3×3 (down 1 and left)
4 3×3 (down 1 and right 1)
4 3×3 (down 1 and right 2)
4 3×3 (down 2 and left)
4 3×3 (down 2 and right 1)
4 3×3 (down 2 and right 2)
4 4×4 (top and left)
2 4×4 (top and right 1)
2 4×4 (top and right 2)
2 4×4 (top and right 3)
2 4×4 (down 1 and left)
1 4×4 (down 1 and right 1)
1 4×4 (down 1 and right 2)
1 4×4 (down 1 and right 3)
2 4×4 (down 2 and left)
1 4×4 (down 2 and right 1)
1 4×4 (down 2 and right 2)
1 4×4 (down 2 and right 3)
2 4×4 (down 3 and left)
1 4×4 (down 3 and right 1)
1 4×4 (down 3 and right 2)
1 4×4 (down 3 and right 3)
1 5×5 (top and left)
1 5×5 (top and right 1)
1 5×5 (top and right 2)
1 5×5 (top and right 3)
1 5×5 (down 1 and left)
1 5×5 (down 1 and right 1)
1 5×5 (down 1 and right 2)
1 5×5 (down 1 and right 3)
1 5×5 (down 2 and left)
1 5×5 (down 2 and right 1)
1 5×5 (down 2 and right 2)
1 5×5 (down 2 and right 3)
1 5×5 (down 3 and left)
1 5×5 (down 3 and right 1)
1 5×5 (down 3 and right 2)
1 5×5 (down 3 and right 3)
1 6×6 (top and left)
1 6×6 (top and right 1)
1 6×6 (top and right 2)
1 6×6 (down 1 and left)
1 6×6 (down 1 and right 1)
1 6×6 (down 1 and right 2)
1 6×6 (down 2 and left)
1 6×6 (down 2 and right 1)
1 6×6 (down 2 and right 2)
1 7×7 (top and left)
1 7×7 (top and right 1)
1 7×7 (down 1 and left)
1 7×7 (down 1 and right 1)
1 8×8
===
204 Total Squares
204
64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204 squares
1×1=64
2×2=49
3×3=36
4×4=25
5×5=16
6×6=9
7×7=4
8×8=1
Therefore, there are 204 squares, of all sizes on a standard 8×8 checkboard
There are 64 individual squares (8*8). Each of those is the lower left corner of a larger 2×2 square, except for those on the topmost or rightmost edges. So there are 49 (7*7) 2×2 squares. Following that pattern there are 36 (6*6) 3×3 squares, 25 4x4s, etc.
The total is 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204
204
May have been around for a long time, but I never saw it before. Number of squares is the sum of the squares, nice!
204 Total
1 8×8
4 7×7
9 6×6
16 5×5
25 4×4
36 3×3
49 2×2
64 1×1
I think I need to go back to the adding charts, jajaja , man, how did I get 221, man oh man
Good work to all of you who found the “sum of squares” solution, or found the result through other means. And to Shawn, I’m glad I could introduce you to this one.
If anyone wants to try a related challenge, try figuring out how many rectangles (including squares) there are.
1296 rectangles including squares
1092 rectangles excluding squares
An easier way to get the result is:
Sum for i=1 to n (i^2) = n(n+1)(2n+1)/6
for n=8: Sum=8*9*17/6=204
For rectangles:
Height 1: 1 x (8+7+6+5+4+3+2+1)
Height 2: 2 x (8+7+6+5+4+3+2+1)
…
Height 8: 8 x (8+7+6+5+4+3+2+1)
Rectangles = (8+7+6+5+4+3+2+1) x (8+7+6+5+4+3+2+1) = 36*36=1296
or
Rectangles= (Sum for i=1 to n (i))^2 = (n(n+1)/2)^2 where n=8
Rectangles=(8*9/2)^2 = 1296
I just like the elegance and the symmetry of the two solutions:
#squares = sum(i^2) for i=1 to n
#rectangles = (sum(i))^2 for i=1 to n
Nicely done, Sirjack and Hex.
@Bobo The Bear, these formulas should please you:
#squares = n(n+1)(2n+1)/6
#rectangles = (n(n+1)/2)^
8×8 =64
7×7 = 49
.
.
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nxn= n^2
sum = 8×8 + 7×7 + 6×6 +. +. +. +1×1 = 204 squares
204+the board =205