Volume 7, No. 2, Fall 2003

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Challenge Answer

by Paul Burney

Our newsletter posed a challenge but didn't provide an answer. Here is the answer and the reasoning behind it:

Answer

2n(n+1), where n is the number of toothpicks on a side.

Reasoning

There are several ways to work out this problem, but here's what I found most easy to follow.

The first thing to do is to look at the case with two toothpicks on a side:

As you inspect the image, think about the problem in two dimensions, that is, horizontally and vertically.

As you do so, you will notice that the number of lines of toothpicks in either direction is n + 1, just as the number of arm rests in an auditorium row is n + 1.

Further, you will notice that each row is made up of n toothpicks.

So, in each direction we need n toothpicks, n + 1 times. Now remember that we will need a set for each direction, so that's:

n(n+1) + n(n+1) = 2n(n+1)

Check

You can check the results of the formula by drawing a picture and comparing it to the results from the formula. A picture gives you an answer of 24. The formula gives:

2(3)(3+1) = 6(4) = 24 check!

Hope that helps!

Paul Burney (pburney@concord.org) is the Concord Consortium Webmaster

The projects described in this newsletter are supported by grants from the National Science Foundation, the U.S. Department of Education, the Noyce Foundation and others. All opinions, findings, and recommendations expressed herein are those of the authors and do not necessarily reflect the views of the funding agencies. Mention of trade names, commercial products or organizations does not imply endorsement.

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