School-Safe Puzzle Games

## 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

### 22 Comments to “Checkboard Trilogy, 1st puzzle”

1. Hex | PUZZLE MASTER | Profile

Size 8: 1×1
Size 7: 2×2
Size n: n^2

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

2. munna | Profile

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

3. JustPlainDan | Profile

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

4. suineg | PUZZLE MASTER | Profile

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)

5. engjs1960 | Profile

204 (64 + 49 + 36 … + 1)

6. APEX.JP | Profile

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

7. Hendy | Profile

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

8. Falwan | Profile

64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204 squares

9. deepside0058 | Profile

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

10. Mashplum | PUZZLE MASTER | Profile

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

11. Shawn | PUZZLE GRANDMASTER | Profile

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!

12. amboutwe | PUZZLE MASTER | Profile

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

13. suineg | PUZZLE MASTER | Profile

I think I need to go back to the adding charts, jajaja , man, how did I get 221, man oh man

14. Bobo The Bear | PUZZLE MASTER | Profile

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.

15. SirJack | Profile

1296 rectangles including squares
1092 rectangles excluding squares

16. Hex | PUZZLE MASTER | Profile

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

17. Hex | PUZZLE MASTER | Profile

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

18. Bobo The Bear | PUZZLE MASTER | Profile

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.

19. Hex | PUZZLE MASTER | Profile

@Bobo The Bear, these formulas should please you:

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

20. Yehuda Groden | Profile

8×8 =64
7×7 = 49
.
.
.
nxn= n^2

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

21. JunzJeahnz | Profile

204+the board =205