## Checkboard Trilogy, 1st puzzle

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*

Hex| PUZZLE MASTER | Profile June 3rd, 2010 - 2:44 amSize 8: 1×1

Size 7: 2×2

Size n: n^2

Total = Sum for i=1 to 8 (i^2) = 204

munna| Profile June 3rd, 2010 - 3:52 am1, 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

JustPlainDan| Profile June 3rd, 2010 - 8:16 amIf 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

suineg| PUZZLE MASTER | Profile June 3rd, 2010 - 8:32 amCool 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)

engjs1960| Profile June 3rd, 2010 - 8:40 am204 (64 + 49 + 36 … + 1)

APEX.JP| Profile June 3rd, 2010 - 8:44 amSquares 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

Hendy| Profile June 3rd, 2010 - 3:03 pmDoing 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

DrJonez007| Profile June 3rd, 2010 - 9:02 pm204

Falwan| Profile June 3rd, 2010 - 10:24 pm64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204 squares

deepside0058| Profile June 4th, 2010 - 3:52 am1×1=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

Mashplum| PUZZLE MASTER | Profile June 4th, 2010 - 10:54 amThere 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

Shawn| PUZZLE GRANDMASTER | Profile June 4th, 2010 - 1:27 pm204

May have been around for a long time, but I never saw it before. Number of squares is the sum of the squares, nice!

amboutwe| PUZZLE MASTER | Profile June 6th, 2010 - 3:41 am204 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

suineg| PUZZLE MASTER | Profile June 8th, 2010 - 9:12 amI think I need to go back to the adding charts, jajaja , man, how did I get 221, man oh man

Bobo The Bear| PUZZLE MASTER | Profile June 8th, 2010 - 12:18 pmGood 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.

SirJack| Profile June 9th, 2010 - 12:25 am1296 rectangles including squares

1092 rectangles excluding squares

Hex| PUZZLE MASTER | Profile June 9th, 2010 - 5:03 amAn 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

Hex| PUZZLE MASTER | Profile June 9th, 2010 - 5:14 amFor 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

Bobo The Bear| PUZZLE MASTER | Profile June 9th, 2010 - 8:36 pmI 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.

Hex| PUZZLE MASTER | Profile June 10th, 2010 - 6:34 am@Bobo The Bear, these formulas should please you:

#squares = n(n+1)(2n+1)/6

#rectangles = (n(n+1)/2)^

Yehuda Groden| Profile October 5th, 2010 - 10:28 am8×8 =64

7×7 = 49

.

.

.

nxn= n^2

sum = 8×8 + 7×7 + 6×6 +. +. +. +1×1 = 204 squares

JunzJeahnz| Profile June 5th, 2011 - 12:09 am204+the board =205