When you cut a deck of cards into 5 piles, there will always be
an even number of piles that contain odd numbers of cards in each pile
and an odd number of piles that contain even numbers of cards in each pile.
Why does this work?
Because 1 or 3 or 5 odd piles would always add up to an odd number of cards, and 2 or 4 even piles would always add up to an even number. An odd number plus an even number equals an odd number of cards, but there is an even number of cards in a deck.So no matter how you divide the cards into 5 piles, there must be either no odd piles at all, 2 odd piles or 4 odd piles. That leaves only 1, 3, or 5 for the number of even piles.
In other words, when there are no odd piles, there are 5 even piles;
when there are 2 odd piles, there are 3 even piles; and
when there are 4 odd piles, there is 1 even pile.