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Building a ruler (Posted on 2008-11-14) Difficulty: 2 of 5
I have an unmarked ruler (AD) of length 6cm. Making two marks in it, one (B) at 1cm from the left end and other (C) at 2cm from the right end, Im able to measure any integer length from 1 to 6 cm:
       A    B             C        D
AB = 1cm / CD = 2cm / BC = 3cm / AC = 4cm / BD = 5cm / AD = 6cm.

If I have an unmarked ruler of length 14cm, what is the minimum number of marks, and where do I have to make them, in order to be able to measure any integer length from 1 to 14cm?

See The Solution Submitted by pcbouhid    
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Catch? | Comment 1 of 10

I have assumed that the marks must go on the cm intervals from 1 .. 13.  I could find no solutions by adding four or fewer marks (between the ends of the ruler).

However, I find 130 distinct solutions for sets of five added integer markings, from 1,2,3,4,9 thru 5,10,11,12,13 (which are symmetrical), which allow all 1..14 integer measures (the ends of the ruler do not need to be marked, but are used for some of the measures -- cf A and D in the text diagram).

Since problems usually are supposed to have unique answers, I am wondering if there is some trick in the wording.

I found the 130 solutions by varying B from 1 to 09, C from B+1 by 1 to 10, D from C+1 by 1 to 11,E from D+1 by 1 to 12, and F from E+1 by 1 to 13.  Measurements were noted from each end of the ruler (A and G) to each marking, and between each pair of added markings.  I have assumed that you do not have the 14cm ruler in addition to the original example 6cm ruler; I see the puzzle is listed under "General" which sometimes allows misleading tricks; hope this is not one of those.

  Posted by ed bottemiller on 2008-11-14 17:35:35
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