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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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re: 16 cm Comment 10 of 10 |
(In reply to 16 cm by monica)

There are 16 ways of doing 16 with 5 marks, not counting reflections:

1   2   3   8  12
1   2   6   9  13
1   2   6  10  13
1   2   7  12  15
1   3   4   9  14
1   3   5   7  15
1   3   5   9  15
1   3   7  11  15
1   3   9  11  15
1   3   9  13  14
1   3  10  11  15
1   4   5   8  14
1   4   7  12  14
1   5   6  13  14
1   6   9  12  14
1   9  10  12  14

The mirror image of monica's is bolded above.

Five marks can also work with a 17 cm ruler:

1   2   3   8  13
1   2   6  10  14
1   2   8  12  14
1   2   8  12  15
1   4  10  12  15
1   8  11  13  15

being the only 6 ways, barring reflections.

For 18, you need 6 marks:

1   2   3   4   5  12
1   2   3   4   8  13
1   2   3   4   9  13
1   2   3   4   9  14
1   2   3   4  10  13
1   2   3   4  10  15
1   2   3   5   8  14
1   2   3   5  11  14
1   2   3   5  11  15
1   2   3   5  11  17
1   2   3   6  10  14
1   2   3   6  11  14
1   2   3   6  11  15
1   2   3   6  11  16
1   2   3   7  10  14
1   2   3   7  10  15
1   2   3   7  11  14
1   2   3   7  11  15
1   2   3   7  12  15
1   2   3   8   9  14
1   2   3   8  10  14
1   2   3   8  11  14
1   2   3   8  11  15
1   2   3   8  12  14
1   2   3   8  12  15
1   2   3   8  12  16
1   2   3   8  13  17
1   2   3   8  14  17
1   2   3   9  10  14
1   2   3   9  11  14
1   2   3   9  12  14
1   2   3   9  13  14
1   2   3   9  13  15
1   2   3   9  13  16
1   2   3   9  13  17
1   2   3  11  12  16
1   2   3  11  13  17
1   2   4   6   8  15
1   2   4   6   8  17
1   2   4   6  10  17

The above list is not necessarily complete, as the program was stopped before completion.

Six marks work all the way through 23-cm rulers:

1   2  11  15  18  21
1   4  10  16  18  21

being the only 2 ways, barring reflections.


  Posted by Charlie on 2008-11-17 13:13:39
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