One unsolved parity problem involves the nucleus of the atom.  Atoms with an even numbers of protons and neutrons tend to be more stable. See Web Elements, Superheavy Elements, and Table of the Nuclides.  The stability of 114 protons and 184 neutrons is still unknown.  Is there a mathematical model that predicts this?

Wei-Hwa Huang made a very nice discovery.  He writes: "I've finally managed to find a convex shape that the thirteen didrafters fit in!  There is a "hidden" parity problem involved -- apparently most of the correct-area convex shapes are (I think) provably impossible -- after I realized this, I tried to construct a few convex shapes that had the "wrong" parity (surprisingly difficult) and the solver managed to find solutions for them!"  These pieces can make a convex shape, as you can see at his site.  (Do look, it's very nice.)  He presents 4 puzzles for this set of pieces.
Puzzle 1.  One solution is missing.  Can you construct it?  The diagram contains at least two hints, but I won't tell you what they are.
Puzzle 2.  What is the parity problem I referred to earlier?
Puzzle 3.  If you solved puzzle 1 without the hints -- what are the hints?
Puzzle 4.  How many convex shapes can be made with the didrafters anyway? (I don't know the answer to this.)

I really like his problem because it introduces a new application for parity.  Parity leads to many excellent proofs, so I decided to write about it.

PARITY

There is a puzzle in the FAQ:  Remove two opposite corners from a chessboard.  Can you cover the remaining 62 squares with dominoes?  Answer: No.  The remaining board has 32 white and 30 black squares, but each domino must cover one black and one white square.

The 56 tiles in a set of Triominoes cannot make a convex shape due to parity.  Joseph DeVincentis explains why.

Sam Loyd invented the 15-14 puzzle.  He offered $1000 to the first person to find a sequence of moves which put at the pieces in order.  By parity, this problem was unsolvable.  To see this, draw a 3x3 grid and place different objects on a1, a2, c1, and c2.  Make moves with the following rule: When one object moves, a different object must move to take it's place.  Moves are thus paired. Now, swap the objects on a1 and a2.  You will find this is possible, but only if the objects on c1 and c2 also swap.

John Conway made a block packing problem.  You must fit three 1x1x3 boxes, thirteen 1x2x4 boxes, one 1x2x2 box, and one 2x2x2 cube into a 5x5x5 box.  The puzzle is hard until you study the parity of each layer, then the problem becomes simple.

Several mazes have been done with parity.  Robert Abbott's Arctic Cave maze is one example.  Adrian Fisher's Veronica maze is another.  In parity mazes, you must find an area that allows you to change your parity.

Can the 35 hexominoes tile a rectangle?  Martin Gardner considered offering $1000 for the first solution, but it's impossible due to parity.  In the figure below, the colored hexominoes are composed of 2 black and 4 white squares if you put them on a checkerboard.  All the other pieces have three squares of each color.  The eleven pieces of unbalanced parity cannot color an equal number of squares of each color.  Since any rectangle with an area of 210 has a balanced parity when you color the squares, the tiling is impossible.  Note how every square has an even number of these unbalanced pieces. (Thanks to Patrick Hamlyn for the figure.)



The sixty-six octiamonds can make a trapezoid with a side of four, can they make one with a side of six?  The twenty blue octiamonds have an unbalanced (3-5) parity when they are placed on a triangular checkerboard grid.  The rest are balanced.  These twenty unbalanced pieces may be matched up two pieces at a time, and therefore grids with parity 0, 4, 8, 12, etc., are probably possible.  A grid with a parity of six, such as the side-six trapezoid, is impossible.  (Thanks to Michael Dowle for this one.)

The Eternity Puzzle is rife with parity issues.  We'll ignore that for now, and instead look at the 59 Hexadudes.  Is there a convex figure they can cover?  Yes, there is!  Patrick Hamlyn just solved it.  The pieces are colored by their parity (balanced, plus two, plus four, plus six).


The 59 hexadudes colored by up/down parity, left/right parity, and east/west parity.


The three parities of ETERNITY -- up/down, left/right, and east/west

For more, see my Eternity Site.