I start this activity by handing out buckets of tiles (any small objects will do) and asking students to take 10. They can sort their ten tiles into any number of piles, and their “prize” is the product of the sizes of their piles — so if they sort them into piles of size 5 and 5; their prize is 5 x 5, and if they sort into piles of sizes 1, 2, 3, and 4, their prize is 1 x 2 x 3 x 4. The goal, of course, is to get the highest possible prize, and then to generalize for other numbers of tiles.
I first saw this problem in a problem section of one of the math journals I get; I can’t remember which. It’s one of my favorites partly because the answer surprised me the first time; it went against my intuition. It’s nice as an activity because it has a definite pattern that students catch onto. The pattern can be challenging for them to describe, and thus is good for working on mathematical notation. Students can work on it in study groups outside of class (I’ve used it as a group oral exam question too).
What’s really great is that with some targeted questions, my students can give a convincing argument as to why their answers are optimal. After they’ve worked on the problem for a while, they understand why their answers are best, and they need a little help to translate that understanding into a solid logical argument.
Extensions of the problem are nice too, although I don’t usually have time in class. We can allow piles of, say, size 2.5, 2.5, 2.5, and 2.5, which gives a score over 39. Or we can experiment with allowing fractional numbers of piles, which is an odd concept, say 2.4 piles of size (10/2.4), for a score of (10/2.4)^2.4. Spreadsheets are useful, and there are calculus connections.
I have a draft of an article on this problem that also makes connections to a finger game that I will post shortly. I am not giving too much away here, as I wouldn’t want you to miss out on trying the problem yourself.