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 Building a ruler (Posted on 2008-11-14)
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, I´m 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 No Rating

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 Difficulty / Unique solutions | Comment 4 of 10 |

Thanks, Charlie, for printing that list.  It is logical that all of the 130 are paired (just mark/label the ruler from right to left instead of left to right, with same relative spacings), giving 65 distinct solutions.  Gamer mentions a "D2" rating, but I am not sure to what that refers (where is it found? what is the scale? who assigns that rating?).  The ratings which appear on the main entry (none yet, for this one) I thought were judgments on the fairness or worthiness of the puzzles, not necessarily sheer difficulty.  (A too-simple puzzle is not worth the time and bother. A puzzle may be hard because it is clever, or because it has not been clearly stated -- quite different issues.

My first approach was a trial and error, which in a few minutes turned up quite a few "almosts" when I tried for a solution with four added marks (i.e. leaving only one of the 14 numbers not capable of measurement). Clearly this made it easy to find a 5 marks solution by just adding that missing spot to the markings, and call it done.  At this point I turned to a computer approach to generate and test all 4-marking sets.  When that yielded none, I expanded the search to 5-markings and turned up bushels of them.  The "number of marks" is ambiguous, since one would not add any markings on the ends of the ruler -- but the text gave letters to those points so that the various line segments would be identified.

 Posted by ed bottemiller on 2008-11-14 19:58:21

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