School-Safe Puzzle Games

Checkboard Trilogy, 1st puzzle

checker board puzzle 1

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


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