Students will learn how to count in base 2, but will learn how to count in base 5 first, as a stepping stone. I think teaching this is necessary, so that students can better understand how base 10 works. Here's what looks to be an excellent approach for teaching base \(5.\) Here's an article for learning the importance of zero in distinguishing between numbers, such as 2035 and 235. Before reading this article, you should know about the base ten system and be able to learn other bases. Here's a more thorough history of zero. There is little overlap between this article and the one mentioned previously. Conclude by giving the following problem, which was taken from Gardner's *Entertaining Mathematical Puzzles:*

A silver prospector was 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 the lady an 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 required 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 on the first day, another inch the second day, then on the third day he could take back the two pieces and give her a solid 3-inch section.

Assuming that 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 needs to cut his silver bar.