Buzz is a old and well-known car travel game.

Here’s the simplest variation. The players begin counting upwards, speaking consecutive integers aloud. If, however, the number is evenly divisible by seven or has a seven in it, you say “buzz” instead of the number. If you make a mistake or hesitate too long, you lose.

The first several buzz numbers are 7, 14, 17, 21, 27, and 28. The game gets tricky when you get to the 70’s and have to count precisely which number is being omitted as the buzzes travel around the vehicle.

Buzz is also a drinking game with obvious consequences for the math-challenged.

While on a recent road trip, I thought of this game and wondered what percentage of integers from 1 to N are buzz numbers. And does that percentage change as N increases?

Before reading further, make an estimate of the percentage of buzz numbers between 0 and 99. How about between 0 and 999? Does the percentage change or stay constant?

I’ll wait….

And you need not put your answer in the form of a question.

Did you play the music?

It’ll help you think (the title of the piece is “Think”…it was composed by Merv Griffin).

Have you made your guesses?

Okay….

It is surprising to most (even to to those of us who formulate math problems instead of counting sheep) that there are 30 buzz numbers between 0 and 99…30%. Of these 30, ten numbers end in 7 (7, 17, 27…), nine begin with 7 (70, 71, 72…don’t count 77 twice!), and eleven more are evenly divisble by 7, but don’t have 7’s in them (14, 21, 28…).

Even more surprising is the discovery that the percentage rises as the interval enlarges. There are 374 buzz numbers between 0 and 999…37.4%.

The percentage continues to rise forever, approaching, but never quite reaching 100%. If that seems counterintuitive, consider this. For any number, the probability that the digit in the ones column isn’t seven is 90%. Same with the digit in the tens column. So, for an integer smaller than 100, the probability that neither column holds a 7 is 90% times 90%, or 81%. That’s 81 non-seven numbers…leaving 19 numbers with 7’s (see above).

For an incredibly large number, one with 100 digits, for example, the calculation would be 90% time 90% times 90%…100 times. The chance that such an integer would be a buzz number would be 99.9974%! And that’s without even considering those numbers divisible by 7.

So, the next time you take a long car trip, by the time you get to Phoenix, the crowd’ll be buzzing.

My head… it is reeling. I think it must be too late for me to wrap my head around this. I’ll have to look again in the AM.