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### Gosh - that's odd!

by Klaus Kohl

Take three successive numbers!
Multiply the first by the third!
Now add 1 to the product!
Take the root!

Variety:

Take some number!
Multiply both numbers!
Take the root!
???

Well, that is a suggestion for a replacement lesson. If the class "hasn't had yet" root extraction it will go this way: multiply two numbers different by two - multiply the number between them by itself - compare the results.

E.g.
5 - 6 - 7 -> 5 x 7 = 35; 6 x 6 = 36
999 - 1000 - 1001 -> 1000 x 1000 = 1000000; 999 x 1001 = 999999
This will always do? Yes, even if one number is zero and it works also with negatives - If they want, you may try fractions...
Well, that is an application of the "third binomic formula" (x-1)(x+1) = x˛ - 1. Who got it first?
And if the two numbers differ not by 2, but e.g. by 20?
Well, then the result will be by 100 smaller than the "appropiate" square number:
2 - 12 - 22 : 2 x 22 = 44; 44 + 100 = 12 x 12
There's the bell?
What, the lesson is finished?