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.)?