March: Invariants


Maksym Fedorchuk, from Boston College, hosted the session for us. We considered ideas involving mathematical invariants – things such as no matter what type of triangle you start with, and how you change it around, its angles will still sum to 180 degrees.  The Law of Conservation of Energy is another such invariant (that the total energy of all objects in an isolated system remains constant no matter what happens to the objects).  Other invariants might not be particularly obvious, and studying what quantities stay the same under certain operations can be a fun and profitable (mathematically, at least!) activity.  Invariants can also be used to show that certain results are impossible – and using invariants is one of the key tools for doing this.

To get our thoughts flowing in this direction, Maksym asked us to imagine an 8 by 8 chessboard with two opposite corners cut out.
(please see attached file if the chessboard image
      doesn't show up!)
Is it possible to cover the chessboard (which now has 64 – 2 = 62 black and white squares on it), with a set of 31 dominoes so that every square is covered?  The dominoes are each 2 by 1 rectangles as shown, and they can be placed either horizontally or vertically as you wish, but they have to align with the chessboard square – i.e. each domino should cover exactly two squares.  Good luck!