A silver seeker could not pay his March rent for an advance. He had a silver bar that was 31 centimeters long, so he made his landlady the following agreement: He told him that he would cut the bar into smaller pieces. On the first day of March I would give the landlady a centimeter of the bar, and every day I would add another centimeter. She would keep the silver in pledge. At the end of the month, the silver seeker hoped to be able to pay the full rent, and she would return the pieces of the silver bar. March has 31 days, so one way to cut the silver was to divide it into 31 parts, each one a centimeter long. but as it was quite laborious to cut it, the silver seeker wanted to fulfill the agreement by cutting it in as few pieces as possible. For example, I could give the landlady one centimeter the first day, another centimeter the second day, and the third day I could give her a part of three centimeters and receive in exchange the two previous parts of a centimeter. Assuming that the bar portions were delivered and returned in this way, see if you can determine the smallest possible number of parts in which the silver seeker must divide its bar.
The silver seeker can fulfill his deal by cutting the 31 centimeter bar into five parts of 1,2,4,8 and 16 centimeters in length. The first day he gives the landlady the piece of 1 centimeter, the next day she gives it back to him and he gives the piece of 2 centimeters; On the third day he returns to give her the piece of a centimeter, on the fourth day she returns both pieces to him and he gives her the piece of silver bar of 4 centimeters. By giving and returning in this way, the silver seeker can add one centimeter per day and thus cover the 31st day of the month.