$13 \cdot 17$ There is a chain made up of 60 rings, each weighing 1 gram. What is the minimum number of rings that need to be cut so that the segments of the chain can be used to make up weights of 1 gram, 2 grams, $\cdots, 59$ grams, 60 grams (the cut rings still weigh 1 gram)?
[Solution] By cutting the 5th, 14th, and 31st rings of the chain, the entire chain is divided into 7 parts, with weights of $1, 1, 1, 4, 8, 16, 29$. It is easy to verify that these can form any integer weight from 1 gram to 60 grams.

When only two rings are cut, the entire chain is divided into 5 parts. A set of 5 elements has only 31 non-empty subsets, so the different weights that can be formed by the 5 parts of the chain do not exceed 31, which of course does not meet the requirements of the problem.
In summary, at least 3 rings need to be cut.