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Table of Odd and Even Parity Bits for Some Binary NumbersMathematically, speaking, parity has to do with being odd and even … with balance.

When binary data is transferred within or between computers, a parity bit allows the receiver to determine if an error has occurred in transit.  In the diagram at left, the original data is grouped eight bits at a time. A ninth parity bit is added to ensure that each parcel of data (including the parity bit) has odd or even parity as required.  The receiver can then deduce whether a bit may have been changed/corrupted during transmission.

Parity can also be thought of in a binary sense … yes/no, black/white, on/off, etc.

Quite a few problems become easy to understand when viewed in this way.

Covering a Modified Chessboard With Dominoes

I challenge you to perform a very simple task ... to cover the squares of a chessboard with dominoes.

But, there is a twist! We first remove two squares from opposite corners. This leaves 62 squares. Theoretically, it should be possible to cover them using 31 dominoes, since each domino covers two adjacent squares. Can you do it?

Image of 8 by 8 Grid With Two Opposite Corners Missing

If you are like most people, you will get or make a board or draw a diagram like the one above and try it out for yourself. After a few attempts, you may begin to believe this is impossible. But ... is it impossible? Is there a clever solution that you overlooked, or is it possible to prove that no solution is possible?

Mathematicians are able to analyse problems like this using a concept of parity ... in this case, the black and white squares of a chessboard. Watch this video and learn how this very clever insight is used to prove whether the problem is solvable, or not!

When you have done that, you might now like to answer these questions: Is it possible to cover the remaining 60 squares if you remove all four corners and only use 30 dominoes? What if you only remove the bottom two corners and use 31 dominoes?

See how simple the puzzles becomes once you have gained this insight? I hope you enjoy the discovery.

By far the best educator I have been subject to in my schooling life. Apart from showing me with the utmost clarity the many concepts of mathematics, Graeme has inspired me to better myself in everything that I do in school and in my life. His dedication to his craft is truly admirable, and this level of dedication to helping the student achieve is something that is extremely hard to find. If you’d want anyone on your side during HSC mathematics, it’d be Graeme.
Harry B (student, 2012)

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