The image at left shows four squares of a standard eight by eight checkerboard with a jumbo-sized
checker in each square. The diameter of the jumbo-sized checker matches the width (and height) of the
checkerboard square. With creative positioning, it is possible to fit more than 64 of these checkers on
an eight by eight checkerboard. What is the maximum number of COMPLETE checkers (i.e. checkers cannot
be cut) that can fit on the checkerboard without going over any edges and without stacking checkers?

**ANSWER**:

*68 (as shown below).*
Here's the math behind it. Let's use

**x** to represent the radius of the jumbo-sized checker. The
diameter of the checker is therefore

**2x** as is the width (and height) of each square on the
checkerboard. The eight by eight checkerboard therefore measures

**16x** by

**16x**. The checkers
in the first column in the image above end at horizontal position

**2x** (the diameter of the checker).
To calculate where the checkers in the second column end, we need to determine the value of

**y** in the
image below:

Following Pythagoras' theorem:

**x²** +

**y²** =

**(2x)²** therefore

**y** =
√

** 3x² ** which is approximately

**1.73x**. If the center of the checkers in each column are

**1.73x** apart, then then the right-hand
most edge of the checkers in each column are also

**1.73x** apart. So:

Column # |
Ending Horizontal Position Of Checker |

1 |
2x |

2 |
2x + 1.73x = 3.73x |

3 |
3.73x + 1.73x = 5.46x |

4 |
5.46x + 1.73x = 7.19x |

5 |
7.19x + 1.73x = 8.92x |

6 |
8.92x + 1.73x = 10.65x |

7 |
10.65x + 1.73x = 12.38x |

8 |
12.38x + 1.73x = 14.11x |

9 |
14.11x + 1.73x = 15.84x |

If we were to add a tenth column of checkers, they would end at horizontal position

**15.84x** +

**1.73x** =

**17.57x** which would extend past the end of the

**16x** by

**16x** checkerboard.
The number of checkers in the nine columns, as shown in the image above, alternates between 8 and 7 as
follows: 8 + 7 + 8 + 7 + 8 + 7 + 8 + 7 + 8 = 68.

Do you have a

suggestion for this puzzle (e.g. something that should
be mentioned/clarified in the question or solution, bug, typo, etc.)?