Solution:

Note that when a link in the center of the chain is cut, three pieces are obtained: a one-link piece and two other pieces. For example, when the third link in a chain consisting of 6 links is cut (see animation), then what is obtained is a one-link piece, a two-link piece and a three-link piece.

The minimum number of cuts needed to be made is 4 for a chain with 96 links.
If the links are numbered serially from 1 to 96, then the cuts would be made on the following links:
6, 17, 38, and 79.

This would result in 4 one-link pieces, one 5-link piece, one 10-link piece, one 20-link piece, one 40-link piece, and one 17-link piece.

To gain a better understanding, consider the scenario in the first few weeks as illustrated in the table below.

Week:	1	2	3	4	5	6	7	8	9	10	11	12
Gold links given:	1	1+1	1+1+1	1+1+1+1	5	5+1	5+1+1	5+1+1+1	5+1+1+1+1	10	10+1	10+1+1

The table above indicates that:
at the end of the fifth week, the 5-link piece is given and the 4 one-link pieces are taken back;
at the end of the tenth week, the 10-link piece is given and the 4 one-link pieces as well as the 5-link piece are taken back.

Food for thought:

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