Math CentralQuandaries & Queries


Question from Shanna, a student:

The paying with silver problem. I understand how to do the problem, but could you please explain how you would use base 2 arithmetic to solve it.


Here is the question that Luther sent us in 1999.

A silver prospector is unable to pay his March rent in advance. He owned a bar of pure silver, 31 inches long, so he made the following arrangement with his landlady. He would cut the bar, he said into smaller pieces. On the first day of March he would give her and inch of the bar, and on each succeeding day he would add another inch to her amount of silver. She would keep this silver as security. At the end of the month, when the prospector expected to be able to pay his rent in full, she would return the pieces to him.

March has 31 days, so one way to cut the bar would be to cut it into 31 sections, each an inch long. But since it requires considerable labor to cut the bar the prospector wished to carry out his agreement with the fewest possible number of pieces. For example, he might give the lady an inch piece in the first day, another inch on the second day, then on the third day take back the two inch segments and give her a three inch piece on the third day. Assuming that the portions of the bar are traded back and forth in this fashion, see if you can determine the smallest number of pieces into which the prospector cut his silver bar. Why?

Claude and I responded to Luther's question but only made passing reference to base 2 arithmetic. The significance of the base 2 comment is that the days in March, the numbers from one to thirty-one, when written in base 2 are

12, 102, 112, 1002, 1012, ..., 111112

Think of these as 5 digit, base 2 numbers

000012, 000102, 000112, 001002, 001012, ..., 111112

Hence the days in March are d5d4d3d2d1 (base 2) where each di is either 0 or 1.

Cut the silver bar into 5 pieces of lengths

1 = 12 inches (first piece)
2 = 102 inches (second piece)
4 = 1002 inches (third piece)
8 = 10002 inches (fourth piece) and
16 = 100002 inches (fifth piece).

For each day in March, write its date in base 2 notation, d5d4d3d2d1 (base 2) and give the landlady the pieces numbered i if di is 1. So for example on March 25 write 25 = 110012 hence give the landlady the fifth piece, the fourth piece, and the first piece.

The argument that four pieces are not enough is in our earlier response.

I hope this helps,

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