Card trick 1
Problem
Try this card trick, it is mathematical. Can you explain (using mathematics) how it works? You might like to consider using some symbolism.
Here are the instructions:
Volunteer chooses 20 cards in the pack to turn face up.
Volunteer then shuffles the pack.
Volunteer removes the top twenty cards without me looking and the remainder are laid on the table.
The twenty cards are handed back to me, without me looking:
I dont know how many cards are reversed in this group I am holding, but it is likely to be less than the number in the rest of the pack. I like things to be balanced so I am going to try to turn some of the cards over so that both piles have the same number face up.
Turn the whole group upside down.
Count out the number which are face up in each pile. (they should both be the same!)
Student Solutions
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.