At Zé's office, there are six pieces of chains with the following number of links: $10, 10, 8, 8, 5$ and 2. He needs to join these pieces to form a circular chain. He spends 1 minute to cut a link and 2 minutes to join it, totaling 3 minutes per link. If he cuts a link at the end of each separate piece, joining the pieces one at a time, he would take $6 \cdot 3 = 18$ minutes. However, since he is in a hurry, he intends to perform this operation in a faster way.

a) Explain how he can form the circular chain in just 15 minutes.

b) Is it possible for him to complete this operation in less than 15 minutes?
Solution

a) One way for him to form the chain in 15 minutes is to initially open all the links of the 5-link piece. In this process, he will spend $5 \cdot 1=5$ minutes. Next, he should use each of these open links between the remaining 5 pieces of chain, using exactly one open link to join two distinct pieces. For this last procedure, he will spend $5 \cdot 2=10$ minutes.

b) The time spent is always a multiple of 3 because once 1 minute is spent to open a link, it is necessary to spend 2 minutes to close it, so the total time per altered link is $1+2=3$ minutes. Since $12=3 \cdot 4$ is the largest multiple of 3 less than 15, if it were possible for him to spend less than 15 minutes, he would have to alter no more than 4 links. Since there are 6 pieces of chain, at least two of them would not have any links altered. To join these two pieces into the larger chain, we would need at least 3 altered links inserted at their ends. Thus, we could use only one link to connect the remaining pieces. Since he will have at least three remaining pieces, this is impossible.