### September 2007: You make the rules

The object of this month's puzzle is to design a ruler that can measure lengths of 1", 2", 3",..., 12". You are given an unmarked 12" ruler and you are allowed to make four marks that divide it into five segments (the marks must be perpendicular to the length of the ruler, as shown below). Describe how to construct such a ruler.

Here's an example to illustrate the idea. Suppose the problem was the same, but you had an unmarked 6" ruler that had to measure 1", 2", 3",..., 6". You can accomplish this by making two marks on the ruler leaving gaps of 1", 3", and 2":

It is obvious how to measure 1", 2", 3", and 6" lengths using this ruler, but also 4"=1"+3" and 5"=3"+2".

Here's an example to illustrate the idea. Suppose the problem was the same, but you had an unmarked 6" ruler that had to measure 1", 2", 3",..., 6". You can accomplish this by making two marks on the ruler leaving gaps of 1", 3", and 2":

It is obvious how to measure 1", 2", 3", and 6" lengths using this ruler, but also 4"=1"+3" and 5"=3"+2".

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