If a bank is creating new paper notes, what sizes should it issue to make transactions easiest for the public? This is one example of an type of problem I've become interested in. It is basically about how to divide a scale into discrete intervals, under various constraints. It seems very abstract, but it actually has a lot to do with real-world design and engineering.

Imagine you're given the task of evaluating the abilities of the individuals in a group -- perhaps in sharp-shooting -- and award ratings like A or F. You might plan a series of challenges of increasing difficulty. What should you do if these turn out to be much harder than you planned -- so that while you expected a mean success rate of 60%, it was actually 20%? An obvious solution is to multiply all scores by 3 to bring up the average. But it turns out that this (and any other linear correction) penalizes the weaker performers. Better alternatives exst.