# Using your Head is Permitted

## October 2008 solution

The trick to solving this riddle is to switch the plus for a minus and to
consider the expression .
Because is clearly less than a third, one can be
sure, without difficult calculations, that is a
very small number, having hundreds of leading zeros after the decimal point.

On the other hand, is easily found to be a whole
number: the even positions in the binomial expansion of both sides are whole
numbers, while the odd positions cancel each other.

The value of is therefore only a very small number
shy of a whole number. Its 48th digit after the decimal point is therefore '9',
as are all other of the first 900 digits and more after the decimal point.

I first encountered this riddle following its publication
here,
but this is not its earliest source. A more complex version appears in
question 5 of
The
29th Annual Virginia Tech Regional Mathematics Contest, and it has surely
been making the rounds even before that.

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