by Rudd Crawford
Once a problem has been solved, there are various ways of playing around with it. One way is simply to look for another solution. The introductory problem about Scott the Painter, for example, has numerous solutions, and in finding them all, one discovers an interesting pattern.
When a problem involves numbers, tweaking the numbers can tease out some interesting results. For instance, Dogs and Ducks states: If a certain number of dogs and ducks have 30 heads and 100 feet, how many of each are there? In solving the problem, we can note that if all 30 animals were ducks, we would have only 60 feet. Replacing a duck with a dog adds 2 feet to the total. We need to add 40 feet, so we try replacing 20 ducks with dogs, giving us 20 dogs and 10 ducks. Then we check: 20 x 4 + 10 x 2 = 100, so we are right — 20 dogs and 10 ducks.
But now that we have solved the problem and are enjoying our success, what if we tweak the numbers? What if there are not 30 animals, but 32? Can we still get 100 feet? Or what if there are not 100 feet, but instead, 106? Can 30 animals have 106 feet? What if there are definitely 20 dogs but a variable number of geese? How many feet are possible then? And so on. Relationships among the numbers become evident — little formulas emerge.
This what-if-not strategy is fully presented in the excellent book, The Art of Problem Posing, by Stephen Brown and Marion Walter, published by the Franklin Press. I recommend it.