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After a deck of playing cards is shuffled by the dealer, it is often given to a player other than the one who performed the shuffle for a procedure called a cut. This is not to be confused with cut cards which are used in casino poker games.
The dealer completes the shuffle, and then sets the cards face-down on the table near the designated player, typically the player to the dealer's right. The player cuts the deck by removing a contiguous range of cards from the deck, and places them toward himself so that the stack of cards to be dealt is closest to the dealer. The simplest form of the cut is done by taking, roughly, the top one-half of the cards, and placing them on the table or a cut card. Either the player cutting or the dealer then completes the cut by placing the remaining bottom portion on top of the cards that have been cut off.
Once the cut is complete, the dealer then picks up the deck, straightens or "squares" it, then deals the cards.
The contiguous section may also be taken from the middle of the deck. This is called "Scarne's cut", though in some settings this is considered poor etiquette or against the rules. A cut involving a very small number of cards, such as taking only the top card (or some cards from the bottom) as a cut, is often acceptable according to rules. The same is true when a player takes every top card save for one on the cut.
During informal card games, the dealer is typically not required to offer the cut, and even if offered, the designated player can decline the request. On the other hand, any player may specifically request to cut the cards before they are dealt. If a cut is requested by a player, it must be granted by the dealer.
In formal player dealt settings, such as in a casino or during a tournament, an offer to cut the deck is mandatory and the designated player must perform the cut, generally by inserting a cut card (a plastic card about the size of a playing card, usually solid-colored) into the deck; the dealer then makes the actual cut at that point in the deck. When the dealer is not a player (i.e. a casino employee), the cut is mandatory and is usually performed by the dealer. In this instance, the deck is cut onto the aforementioned cut card, and the cut completed; this prevents players from seeing the bottom card of the deck.
A cut should always be completed with one hand to limit possibility of a false cut.
Scarne's cut was developed by John Scarne during WWII to help protect servicemen against cheating by unscrupulous dealers. First one pulls out a portion of the middle of the stack and places it back on top of the deck; one then performs a regular cut described earlier.
The practice of cutting is primarily a method of reducing the likelihood of someone cheating by manipulating the order of cards to gain advantage. Even if the dealer does not plan on cheating, cutting will prevent suspicions, thus many rules require it. Some players also consider the cut to be lucky.
It can be demonstrated that multiple top-to-bottom (non-Scarne's) cuts are equivalent to some single cut. In fact, knowing the size of the deck and the size of the cuts, the formula for the composite single cut is given as the sum of the sizes of the cuts modulo the size of the deck. For example, in a 10 card deck, if a 7 card cut and a 4 card cut are made, that is, 7 cards are moved from the top of the deck to the bottom and then the resulting top 4 cards are also moved to the bottom, then those two consecutive cuts are equivalent to a cut the size of (7 + 4 = 11 (mod 10)) = 1. The deck will be in the order (2,3,...,10,1).
A false cut is a move used either in magic, or for cheating when playing card games. It appears to be a real cut, but leaves the deck in the same order as when it began. More sophisticated versions may make specific desired changes to the deck's order, while still appearing to be an innocuous normal cut.
There are many ways to accomplish a false cut, involving misdirection or using complex moves to conceal the real result.
As a game
Cutting cards is usually a prelude to a game, but it can be a game unto itself. Each player, in turn, removes a selection of cards from the top and reveals the bottom card to all the players, and then replaces the cards in the original position. Whoever has revealed the highest (or sometimes lowest) card is the winner. This is often used in an informal setting, much like flipping coins; it is also sometimes used to determine who will play first in a card game.
As a joke
The command to "cut the cards", following by someone literally chopping the deck in half with an axe, is a none-too-subtle gag that has been used many times in popular media, going back to at least the vaudeville days. Examples include Harpo Marx in Horse Feathers, Curly Howard in Ants in the Pantry, and Bugs Bunny in Bugs Bunny Rides Again.