# An Auction

Question:

A woman wants to buy something at an auction where you bid grams of gold instead of money. She owns a length of gold chain (with 2 ends) made of 23 interlocking loops, each weighing 1 gram.
She wants to go to a jeweler before the auction to cut the minimum number of loops that would allow her to pay any sum from 1 to 23. For example, she could pay a 13 gram price with a 12 link chain and a single link. After much thought, she figures out a way to do it by cutting just 2 of the loops in the chain.
How many loops are in the pieces of chains that she has after the 2 cuts?

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Solution: She will have five pieces of chains after the 2 cuts. The chains will have loops of 1, 1, 3, 6 and 12.

The chain is made up of interlocking loops. So when you make 1 cut in the chain in the middle, you would end up with 2 chains and 1 cut loop. So 2 cuts in the middle would leave you with 3 chains and 2 cut single loops.
So the jeweler makes one cut on the 4th loop making a 1 cut loop, a 3 loop chain, and a 19 loop chain. Then a cut on the 7th loop of the 19 loop chain leaving a 1 cut loop, a 6 loop chain, and a 12 loop chain. So at the end, she has 1 cut loop, another 1 cut loop, a 3 loop chain, a 6 loop chain, and a 12 loop chain.\