Sunday, January 31, 2016

NP-Complete Problems as Repeatable Math Puzzles

Repeatable Puzzles

 In the 3 years I worked at Matific, I was involved in designing all of the mini-games (we called them 'episodes') produced by the company. This meant that I spent a non-negligible part of my time on the hunt for math puzzles that are
  • interesting and rewarding to solve
  • rely only on elementary school math knowledge
  • worth playing over and over, with respect to the above
Most of the really cool puzzles, such as determining which - out of 12 coins - was fake, using a balance scale and only 3 weighings, fall short of the "worth repeating" notion, as there's no real pleasure or challenge in solving them a second time. So googling "Math Puzzles" and reading puzzle books, though fun, was often of little use.

As time went by, and development progressed, puzzles turned out to be only a small part of what we did. A typical episode was rather required to help in understanding some deeper principal of a previously taught skill, via hands-on experience, which I think we managed to do well, and that's a topic for another post. 

Nevertheless, there were some puzzles in the mix, for example: a game where the user is given 4-5 numbers (e.g. 2, 3, 4, 7), and is asked to get to a target number (e.g. 23) using the some of the four operations and each of the initial numbers (e.g. 4/2 + 3*7).

Another puzzle, for 1st graders, asked the student to arrange toys on shelves, subject to some constraints, e.g. - the doll must be above the teddybear, the car has to be to the right of the dinosaur, and so on.

A third puzzle was simply variants of a well known water-pouring puzzle: one has 2-3 containers, each of different volume (e.g. 5L and 3L) and the objective is to obtain another quantity (e.g. 4L) via pouring and filling.

Starting With the Solution

 As we designed more of those repeatable puzzles, a few common traits emerged:

  •  they involved a more than one skill. The pouring puzzles, for example, involves addition, subtraction, understanding of volume, and strategizing.
  •  we had to generate problems with well-tuned difficulty levels.
Let me focus on that last one: When one designs a puzzle, the first instinct may be to randomize the problem (e.g. how much water in each container? what is the target quantity?) and then write a solver that solves it, but that is rarely the right thing to do, for two reasons: 1) This leaves the difficulty level to chance, 2) Sometimes the problem has no quick solver, and the solver will needs to go over all possibilites to find the solution, or reach a conclusion that it has no solution, in which case a new problem has to be generated at random and checked and so on. Our standard strategy, which applied to most of the cases, was starting from the solution. 

For example, choose 4 random numbers in some range, e.g. 2, 5, 9, 11, and now choose operations at random, e.g. *, +, -, which gets 2 * 5 + 9 - 11 = 8. And so we present the user with the numbers 2, 5, 9, 11, setting 8 to be the target number. This approach has its drawbacks, such as producing problems using a certain solution while a simpler one exists, but they are less severe, product-wise.
This "start from the solution" approach took us a while to figure out, in which time we tried the naive approach, of randomizing a problem and running a solver. After a few of these, I noticed that we often realize that the problem is NP-hard, or in some cases assume it is so, not being sure how to prove it. A solver for such puzzles is problematic and can't scale to larger instances of the problem, especially seeing how the code had to be very light-weight to run on a browser. Lastly, a crucial part of programming a puzzle is checking the user's solution, which was almost always considerably easier than generating the problem. And so at some point it became clear to me that many of our so-called puzzles, were simply instances of NP-complete problems.
This was neat because it gave us something to draw inspiration from, in addition to poring over puzzle books.

Some Thoughts 

 I think that what makes NP-complete problems such useful puzzles is that they somehow seem to come from the real world, at least more than those in harder complexity classes, which means many of them can be explained to a layman in a minute. The NP-completeness also means that there's no trick that the user learns once, and is then bored by subsequent puzzles that are solved using that same trick. Being in NP also means that solutions can be checked easily, so the software can check the user's solution, but also, when the user fails - and the software shows a correct solution, the user can see very quickly that it is correct. Lastly, they encourage a heuristic cross-topical approach to problem solving, which is to say there's no title saying "this is a subtraction problem, solve it using subtraction". Both the method and the tools used by the user (or student) are not part of the problem's definition, and that is how math usually presents itself in real life.