Checkerboards and Fences
Old checkerboard challenge
How many squares are there on an 8x8 checkerboard, counting not only the 64 1x1 squares, but also the 2x2 squares, 3x3 squares, etc.? What about a 101x101 checkerboard? How many of those squares contain the little square in the center?
On an 8x8 checkerboard, there are 64=82 1x1 squares. There are just 72 (overlapping) 2x2 squares, since they cannot start in the last row or column. There are 62 3x3 squares and so on down to 12=1 8x8 square. The total is: 82 + 72 + 62 + 52 + 42 + 32 + 22 + 12 = 204.
Similarly on a 101x101 board the grand total is: 1012 + 1002 + ... + 22 + 12 = 348,551.
Some readers added this sum one term at a time. Others derived or looked up a formula for the sum of the first n squares: n2 + (n-1)2 + ... + 12 = n(n+1/2)(n+1)/3. Plugging in n=8 yields 204; n=101 yields 348,551. The number of squares on the 101x101 board containing the center square is: 12 + 22 + ... + 502 + 512 + 502 + ... + 22 + 12 = 88,451.
(Best answers from William Bond, Reed Burkhart, Gary De Young, Chuck Gahr, Chuck Munyon, Ethan Plunkett, Luke Somers, Mike Soskis, Clinton Staley.)
The rectangular enclosure of a given area that uses the least fencing is a square. What is the most efficient shape for two identical rectangular pens, adjacent along their length?
Not two squares, but two rectangles that are a third longer than wide, to take full advantage of the common wall that serves both. For example, if each is to have a 12-sq.-ft. area, each would be 4 feet by 3 feet.
The best dimensions can be found by trial and error or computed as an easy calculus problem.
Double pen challenge
What is the most efficient shape for two identical adjacent pens if any shapes are allowed, not just rectangles?
Send answers, comments, and new questions to:
Bronfman Science Center
Williamstown, MA 01267
or by e-mail to:
The best submissions will receive a copy of the book "Flatland," which explains higher dimensions. The answer to the challenge question and winners will be announced in the next column in two weeks.