Hello, dear reader! If you are a new reader, welcome! (I know there is at least one of you—thanks to commenter sg211 for prompting me to post again. =)
I must apologize for dropping the ball on writing up the solution to part II of the powers of two challenge from a couple months ago, and in a broader sense wrapping up my ongoing series on Recounting the Rationals. One reader even wrote me an e-mail with a friendly reminder to write up the solution! Well, I haven’t forgotten about it, and I promise to get to it soon.
In the meantime, here’s an interesting problem for you. I have a little clear plastic piano-shaped thing that sorts US coins—you put coins in the top, and it sorts them into stacks by type. I went to put some coins in the other day and noticed that there were a ton of quarters in it, but almost no nickels, which made me wonder whether there was a reason for that, or whether it was just some sort of fluke. So, the question for you is: was it a fluke? Or would you expect me to have fewer nickels than quarters in my coin-sorting piano?
[For any non-US readers who don't know, the values of US coins are 25 (quarters), 10 (dimes), 5 (nickels), and 1 cents (pennies). Actually, there are also coins worth 50 (half-dollar) and 100 cents (dollar), but those are far less common.]
To be a little more precise, you can make the following assumptions, all of which are, of course, completely false in the real world, but hey, this is a math problem. =)
- I started with a completely empty coin-sorting piano.
- I never use any coins in making purchases, but always pay with bills only.
- The cents portion of the prices of things that I buy are random and uniformly distributed between 0 and 99; in other words, for any particular number n between 0 and 99, there is a 1/100 chance that the price of something I buy will be some number of dollars plus exactly n cents.
- Cashiers always give me change in quarters, dimes, nickels, and pennies (no dollars or half-dollar coins), using the fewest number of coins possible. (For example if I am owed 16 cents in change, I would get one dime, one nickel, and one penny, not three nickels and a penny or anything like that.)
As usual, post comments, questions, and/or solutions in the comments (don’t peek if you haven’t tried solving it first! =)