Tuesday, July 20, 2010

Solving Number Puzzles with Matrices

The Discovery Education Classroom Resources has some great classroom tools. One of my favorites is Puzzle Master. It allows you to create custom puzzles like word searches or cryptograms. I used the Number Blocks puzzle creator for a linear algebra project on overdetermined and underdetermined systems of equations.

An example of a number block puzzle created using Puzzle Master is given above. The numbers to the right an bottom are the sums of various rows, colums and diagonals of the 3-by-3 grid of numbers. The main diagonal sums to 14; the off diagonal sums to 17. The sum of each row is to the right, and the sum of the columns is at the bottom. In this example, there are 6 unknowns and 8 equations, and so this is an overdetermined but consistent system. Puzzle Master will not create a puzzle that does not have a solution.

Questions:  We can create a system of equations for this number puzzle as shown above. There are a lot of questions that can be asked about these number puzzles and the systems that can solve them.
  • Given an n-by-n grid of numbers, how many equations are needed to solve the puzzle?
  • What determines how many unknows there are?
  • Given an n-by-n grid of numbers, how many unknowns must there be for the system to be overdetermined? Underdetermined? Square?
  • Are there equations in the system below that can be removed so that the system is not overdetermined but has an equivalent solution? If so, how many and which equations can be removed.
  • If a number puzzle is underdetermined, how can a solution be found?
  • Is it possible for a number puzzle to have exactly two different solutions? Exactly five different solutions? Explain.
  • Is it possible for a number puzzle to have infinitely many solutions? Describe a puzzle for which this would be true, and explain how to describe the infinitely many solutions.
  • Puzzle Master won't create a puzzle with no solution, but can you create a puzzle that has no solution and is overdetermined? Underdetermined? Square?
  • Create a puzzle using Puzzle Master that would have infinitely many solutions. Can you make a conjecture on how Puzzle Master chose the solution that it gives?

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