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Shelf Tidying
The Problem:
The twenty volumes of Star Trek videos are slightly out of order…and it’s your job to
tidy them up.
We’ll count “taking a video off the shelf, pushing some of the others to one side, and
putting the first video back onto the shelf” as a single move that takes about, say, 5
seconds.
What is the quickest time you could get these twenty volumes back into their correct
order, reading 1 to 20 from left to right?
Can you prove it’s the quickest time possible?
Hint:
Are any of the Volumes in the right order?
Solution:
The quickest time possible is 60 seconds.
Reading from left to right, we can recognize several sequences of numbers already
in the correct order with respect to each other…
13 8 3 5 12 16 6 2 1 9 17 4 19 20 10 11 7 14 18 15
i.e.
8-12-16-17-19-20
13 8 3 5 12 16 6 2 1 9 17 4 19 20 10 11 7 14 18 15
i.e.
2-9-10-11-14-18
The longest such sequence we can find (of length 8) begins with Volume 3…
13 8 3 5 12 16 6 2 1 9 17 4 19 20 10 11 7 14 18 15
3-5-6-9-10-11-14-15
We could use Volume 18 instead of Volume 15, of course.
This then leaves the other 12 ‘out-of-order’ videos that need moving.
Hence the least number of ‘moves’ needed is 12 , and the quickest time for the task
will be 12 x 5 = 60 seconds.