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 DAB = 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?
(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.
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Posted by Charlie
on 2008-11-17 13:13:39 |
re: 16 cm
