Quentin L. Meunier

Problem 1009

A paper manufacturer has chains of paper clips attached to each other.

During a "manipulation", he detaches a paper clip at both ends and gets this paper clip as well as two new smaller chains.

1A. From a chain of 63 paperclips, how many manipulations will he have to perform at least, to be able to provide, without further manipulation, any number of paper clips between 1 and 63?
1B. What is the longest initial string that allows, in 8 manipulations, to provide any number of paperclips between 1 and the length of the chain?

The paper manufacturer receives a chain of 200 paper clips of three different colors: the first is red, the second blue, the third green. He notes that by removing one out of four paper clip (starting from the fourth), the remaining 50 paper clips removed are identical to the first 50 paper clips in the initial string and the remaining 150 are identical to the first 150 paper clips of the initial chain.

2A, 2B, 2C. What is the number of red (2A), blue (2B) and green (2C) paper clips in the chain?

For question 1B, the following program makes it possible to check that a given division makes it possible to reach all the numbers of paper clips. An exploration of all the divisions obtained by the various possible manipulations is also possible but explodes very quickly in the number of configurations.

For question 2, the following program implements the rules and calls them iteratively until all the colors of the string are known.

1A. From 63 paper clips, we obtain all the numbers from 1 to 63 with at least 3 manipulations.
1B. With 8 manipulations, the longest possible chain is 4 607 paper clips.
2A, 2B, 2C : In the chain there are 120 red paper clips, 46 blue and 34 green.