>  I suppose the kid's reasoning was something like: "Hmm, actually if I
> take the *average* of all the numbers, and multiply it by the amount of
> numbers, I should get the correct result, because summing the average
> 100 times should be the same as summing all the original 100 numbers.
> Now, what is the average of all the numbers between 1 and 100? It must
> be (100+1)/2. Now multiply that by the amount of numbers, ie. 100, and
> we have the answer."

The version I heard was that he imagined the numbers written 1-100, and then 
the same numbers written 100-1 underneath.  Like this:

  1  2  3  4  5  6 ...
100 99 98 97 96 95 ...

Now if you sum up both of those you should get twice the answer right?  But 
first he summed each column of two numbers to get:

101 101 101 101 101 ...

Then of course it's trivial.

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