Necessary Versus Sufficient

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It's easy to get carried away with showing that a particular condition or feature is necessary, and then automatically assume that it is also sufficient. It doesn't always follow.

Cartoon sketch of the
bridges in Koenigsberg
Take the famous case of the bridges in Koenigsberg.

Euler showed that you can't walk over each bridge exactly once because it didn't have the appropriate necessary condition. He showed that to be able to walk over the bridges exactly once it is necessary that the number of places with an odd number of bridges is either 0 or 2.

However, it seems that when people use this as an example they automatically to assume that if each place does have an even number of entry/exit points then it therefore can be drawn without lifting your pen off the paper.

That's not true!!

Can you see why not?

Another place where the question arises is given on the Dominoes Unlimited page. There we see that it's impossible to cover a chessboard with dominoes if opposite corners are removed. The reasoning is simple. If it can be covered then we must have the same number of black and white squares.

But is that sufficient?


So don't get carried away about proving just one direction.

You don't? Are you sure?

Tell me then: Just what is Pythagoras' theorem?


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