by Michael Cain
(submitted by request)
A recent comment thread headed off into a discussion of the attractions of games and puzzles that involve combinatorial search, like Wordle or Sudoku or Freecell. Here's an example of a combinatorial puzzle. My daughter brought this home from math class when she was in eighth grade (long ago).
On the way home from work I stopped at the corner convenience store to pick up four items. The checkout clerk did things on the register and told me "$7.11, please."
"That seems too much. How did you calculate that?" I asked.
"I multiplied the four prices together."
"Aren't you supposed to add the prices?"
"Oh, right." After a moment he said, "Still $7.11."What were the prices of the four items?
She told me the math teacher was explaining a technique he called guess and check: guess at the answer and check to see if it's correct. She thought it was stupid and clearly expected me to think the same. She was surprised when I said, "Cool! There's a whole bunch of neat math in there!" We talked about problems where you had to choose from a set of possibilities and had to find the right combination to solve the problem. That you often needed to find a clever strategy so you could find the right combination in a reasonable amount of time. We played around with this particular problem some, but didn't guess the right answer before it got tiresome. (No one else in the class guessed the right answer either.)
Some years after that I was working at an applied research lab that did lunch-time technical talks. I was asked to do one that had some math, some entertainment value, and that most of the staff would be able to follow. My recollection of the talk about the 7-11 problem is reproduced below the fold.
Oh, and open thread, because why not?