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Enumerative Solutions - The First Matchstick Problem

Problem: How many matches do you need to construct a sequence of n squares (as shown below):

Look through the sequence of building squares:

Let us build a table indicating how many matches are need to build each set of squares:

Number of squares
Number of matchsticks
1
4
2
7
3
10
4
13
5
6
7

Fill in the missing values. Click outside the box once you have entered each value to check the correctness of your entry.

It should be fairly obvious that each time we add a new box, we also add 3 new matchsticks. Thus we can conclude that the general formula for calculating the number of matchsticks for nboxes is:

4+3(n-1)

That is, 3n+1

This process of enumerating the solutions and then deriving a general formula is the process of induction. You can try another example by looking at the number of matchsticks needed to create matrices of boxes.

Last updated 2001/02/27