A Bone to the Math People

Since my recent posts have been political and personal, I thought I'd put up a mathy post so all you math people don't desert me.  Maybe you can answer a question for me.  I once heard a discussion of the following question:  "Is there anything special about the number 2?"

One answer, "It's the only even prime", was labeled not a real answer, since that specialness only resides in the fact that we have a name ("even") for numbers divisible by 2.  If we had a name for numbers divisible by 3 (say, "threeven"), then 3 would be special, too, since it would be the only threeven prime.

Other answers depended on some group theory that was beyond my limited understanding of that subject.  But I have another answer.  The only problem is, I'm not sure if my answer just boils down to "it's the smallest integer greater than 1".  If it boils down to something deeper, please tell me!

It stems from a problem that I like to give 7th- and 8th-grade students:

Crazy Al owns a dollar store that isn't like any other dollar store.  It's not that everything costs one dollar, it's that everything costs at most one dollar.  In fact, every possible price (from 1¢ to $1.00) is represented by at least one item in the store.  Furthermore, Crazy Al has a big sign behind the counter that says "I will not give change, EVER!!"  So if you want to buy an item and you don't want to get ripped off, you have to bring exact change.  One day you need somthing in Crazy Al's store, but you don't remember how much it costs.  So you have to bring enough coins to be sure that you can pay for any item in the store with exact change.  With (common) American coins of 1¢, 5¢, 10¢, and 25¢, what's the smallest number of coins you could bring to be sure you could have exact change for any item?  What about with EU coins of 1¢, 2¢, 5¢, 10¢, 20¢, 50¢?  What if you could design a coinage system just for this purpose that was the most efficient way to guarantee exact change for items up to $1.00?

The answer, of course, is to have coins of 1¢, 2¢, 4¢, 8¢, 16¢, 32¢, and 64¢.  The powers of 2.  This seems clearly related to the fact that if you're going to build/write integers, 2 is the smallest possible base (which is because it's the smallest integer larger than 1).  But is this greatest efficiency a sign of anything deeper?  Is 2 being any more special here?

By the way, this is even a great party problem for adults...when they get into "I don't understand how math could be fun" mode with you, this problem (in 2 stages:  smallest number of coins, then devising the best possible coinage system) can often demonstrate how math doesn't have to be abstract to be interesting.

Okay, I'm off to my grandmother's 90th birthday party in the morning.  I've heard that birthdays are good for you--the more you have, the longer you live.

Comments

[vlorbik]

this appears to be a very interesting question (but threatens to become a pseudo-question: if one were to try to discuss it "seriously", it seems very likely that people would commit themselves to "taking sides" -- e.g., "your answer is actually just another *consequence* of 2 being the smallest prime [or whatever property one has singled out as the most important one]).

anyhow. i'm with whoever it was that started his philosophy with "draw a distinction": the property of being one more than 1 is not to be sneezed at. it's the reason binary arithmetic is the very lifeblood of "digital" computing (how odd, excuse me, strange, that "digital" suggests "counting on one's fingers" . . .).

keep 'em coming.

[RobinFiveWords]

From a linguistic point of view, there is definitely something special about the number 2. [I'm relating a story that I read in _The History of Pi_ by Petr Beckman.]

Today we may perceive quantity as either singular or plural, meaning "one" and "more than one," respectively. In man's history, though, there were originally separate concepts for "one," "two," and "more than two," and our languages contain remnants of this. In English, for example, the term for the reciprocal of two is half, but the term for the reciprocal of every number greater than two is the ordinal number derived from the name for that number.

[discount cigarettes]

Crazy Al owns a dollar store that isn't like any other dollar store. It's not that everything costs one dollar, it's that everything costs at most one dollar.