Last week’s AlgPoW, Don’t Be Square, was (I think) a simple problem hidden behind lots of complications. It is a perfect problem to think about different ways to look at a hard situation and think about how to use the Solve a Simpler Problem strategy to eventually crack the whole, hard problem.
The PoW went something like this:
√((3/2)*(4/3)*(5/4)*…*(a/b)) = 10√10.
If the whole problem were completely simplified, it might look something like this:
a/2 = 1000
b = a – 1
Find a and b.
Easy, right? But that’s not the problem we gave, on purpose. We gave a problem where the solver’s job is to look at a mess and say to themselves,
- What’s making this problem hard?
- What patterns can I find and write more compactly?
- Are there other, simpler ways to write the same thing?
- What can I learn from replacing hard numbers with easier ones?
- Could I do the problem if…?
- What can I learn from doing a few sample of this long pattern?
- What happens if I try specific examples?
Those are all questions that are really valuable in all kinds of hard problem situations, whether the problem is hard because someone made it tricky for you, or because it’s one you stumbled across in daily life (like the problem I was stumped on for a while: how many times can 36 people in groups of 6 change groups before someone has to be in a group with someone they were with before?).
So, what happens when students ask and answer these questions? Here are some examples from the Don’t Be Square PoW.