Bilbo has a gold chain with 20 links.  He is staying in town for 20 days at an inn and wishes to pay each day with one gold link.  When he cuts the chain, the remaining links looks like the diagram below: 

The diagram shows 10 links but there are 20 on Bilbo’s chain.

If you cut at the 10^th link, you will have 10 links on one side, one link in between and 9 links on the other side.

He wants to cut the least number of links possible and pay just one link per day.  The innkeeper does not want to take the 20 links up front and does not want more than his share per day.  But, he is willing to trade Bilbo the links to make even. 

For example, if Bilbo pays with the extra link on day one, he can give a chain with two links on day two and get the extra link back.

Thus, help Bilbo out and tell him what is the least number of cuts he needs to make.  Also, make a diagram describing how the transactions go each day.