Choose any three by three square of dates on a calendar page...
Can you explain how this card trick works?
Take any whole number between 1 and 999, add the squares of the
digits to get a new number. Make some conjectures about what
happens in general.
When the pack has been shuffled and then split, let us assume
that there are x cards face up in the pile of 20. This
being the case, there must be 20 - x cards face up in the
second pile. When the pile of 20 is turned upside down there will
be 20 - (20 - x) = x cards face up in the pile.
Therefore both piles of cards will have the same number of cards (
x ) face up.
Of course the trick works just as well if the volunteer chooses
25 cards or any other number of cards. You might like to convince
yourself that this is so by rewriting the above solution replacing
20 by 25.