# The payment to the innkeeper

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A humble peasant, goes on the back of his faithful mare to ask his beloved Mary in marriage. During the long trip his mount suffers a mishap that prevents him from continuing with the trip. Luckily, it is near the inn where you can get another horse. The innkeeper tells him that he will not have horses until after seven moons, so the poor farmer has no choice but to stay in the inn until the animal arrives. Carlos has 10 silver coins, enough to buy the horse but not enough to pay for the accommodation, so he proposes the following to the innkeeper.

I have a chain of 7 silver links. At the end of my stay I will deliver it as payment for the accommodation.

The innkeeper, to whom the farmer does not inspire any confidence, replies. You will give me a link in the chain every day you stay at my inn.

The peasant, who has a reputation as a low worker, thinks of the best way to cut the links in the chain to meet the requests of the innkeeper making the minimum number of cuts possible.

What is the minimum number of cuts that the farmer must make to the chain to meet the requests of the innkeeper and how should he deliver them?

#### Solution

It is possible to meet the requests of the innkeeper with a single cut in the chain as follows:

If we cut the third link in the chain, we will have a loose link plus a chain with two links and another with four links:

• The first day we deliver a link to the innkeeper.
• On the second day we deliver the chain with two links and the innkeeper will be forced to return the link we gave him the previous day, so that we finally satisfy the payment of a link for that day of accommodation.
• On the third day we deliver the loose link that returned us the day before.
• On the fourth day we deliver the chain with four links and the innkeeper will be forced to return the loose link plus the chain of two links.
• On the fifth day we will deliver the loose link that returned us the day before.
• On the sixth day we deliver the chain of two links and you will be forced to return the loose link that we delivered the previous day.
• Finally on the seventh day we will deliver the loose link and we will have fulfilled the requirements of the innkeeper to deliver a link every day making a single cut to the chain.