One of the finest of all mathematical magic tricks, not well known even to card magicians, was invented by Howard Adams.   It was first published in his now rare 1984 booklet titled "OICUFESP" (as in "Oh, I see you have ESP").

I urge you to get a deck of cards and astonish yourself by following these simple instructions.   First, remove from the deck a set of five pairs of "mated" cards.   (A mate to a card is a card with the same value and color, i.e., the mate of the queen of hearts is the queen of diamonds.)   Call the five cards ABCDE and their mates abcde.   Arrange them in the order ABCDEabcde and place the packet of 10 cards facedown on a table.   You can then cut this packet of cards as many times as you want.   (Cutting a packet means lifting some cards off the top, thereby separating the packet into two parts, and then transposing the two parts as you put them back together, always keeping the facedown orientation of all the cards.)   Then split the packet in half (i.e., cut it into two sets of five cards each) and place the halves facedown on a table, side by side.   Then turn the five-card pile on the right faceup.

You are now going to spell the words in the phrase LAST TWO CARDS MATCH in the following way.   Pick up either pile (you can let someone else choose which one) and spell the letter L by moving the top card of the pile to the bottom, and then putting the pile back on the table without turning it over.   Likewise spell the letter A of LAST by once again picking a pile at random, moving its top card to the bottom, and putting the pile back on the table, keeping the faceup or facedown orientation of the cards.   The spellings of S and T are done in the same way.   After the first word is completed, we still have two piles, one faceup and the other facedown.   Now remove the top card from each pile and set them aside, side by side, at some vacant spot on the table.   One will be facedown and the other will be faceup.

Repeat the process with the remaining piles of four by spelling the word TWO, picking a pile at random for each letter.   After the word is spelled out, remove the top card from each pile and set them aside as before, side by side, below the previously removed pair of cards.   Proceed by spelling CARDS and MATCH with the remaining piles, again removing the top cards after completing each word and setting them aside.

You will now have two columns of cards (made up of the four sets of removed cards), plus the two cards that are the last two cards left in your starting piles.   Focus on those last two cards, and turn over the one that's facedown.
Surprise-it matches the other card!
,br> But that's not all.   The trick has a second, even greater climax.   Turn over the remaining four facedown cards.   Each one will match the faceup card it is paired with!

Maybe you can figure out why the trick works.   It is based on modular arithmetic.   Once you have figured out why, you may be ready for the challenge of devising a similar magic formula that will work for larger sets of cards, or even for the entire deck!