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The seven eslabones

It is an exact change piece disguised as a chain: minimal, clean and with that rare satisfaction of seeing 1, 2 and 4 appear where there seemed to be only seven alike.

A blacksmith agrees to pay a laborer with a small gold chain made up of 7 closed links. The deal is this: for 7 days, you'll need to hand him exactly one gold link per day.

You can take back links already delivered to make change, but you want to make as few cuts in the chain as possible. How many cuts do you need at least?

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Hints

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Don't think about seven loose links. Think about what piece sizes would be best for you to build any number from 1 to 7.
If you could have pieces of 1, 2 and 4, you could make any number between 1 and 7.
A single cut, made in the right place, is enough to turn the chain into pieces of those sizes.

Solution

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The idea is to create pieces that allow you to represent any quantity from 1 to 7 also using returns as change. The ideal pieces are sizes 1, 2 and 4, because any number from 1 to 7 can be formed with them. This is achieved with a single cut, opening the third link counted from one end of the chain. By doing so, the chain is converted into three usable pieces:

a loose link of 1
a section of 2 links
a section of 4 links From there, the daily payment works like this:

day 1: delivery 1
day 2: recover 1 and deliver 2
day 3: delivery plus 1
day 4: recover 1 and 2, and deliver 4
day 5: delivery plus 1
day 6: recover 1 and deliver 2
day 7: delivery plus 1

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