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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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Some Thoughts just 2 remarks | Comment 6 of 10 |

Marking  1  3    6  and 10 allows repesentation of all distances,

except 12.

Theoretically 6 marks could cover 15 distances ==> C(6,2)=15.

However not all the pairs define different distances,
D(0,3)= D(3,6)   and D(6.10) =D(10,14)  so we are left with partial coverage only, (15-2=13) ==>12 is missing.

REM1 : My 4 marks are triangular numbers, but for the ruler
of 15 cm (next triangular number) the situation does not improve

REM2 : The fact that there is no possible 4tuple of different numbers between 1 and 13 inclusive can be be easily proven  by checking exhaustively all possible combinations

  Posted by Ady TZIDON on 2008-11-15 11:11:26
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