Card Games: War

This problem is based on the card game War .

Rules: The deck is divided evenly among the players (usually two). Both players must simultaneously turn over the top card in their pack, and the player with the higher card takes both cards played and places them at the bottom of his/her deck.

If the two cards played are of equal value, both players put three more cards from their stack on top of their pile face down, and then another card face-up. The owner of the higher face-up card wins all the $10$ cards on the table. If the face-up cards are again equal then the battle repeats with another set of face-down/up cards. This repeats until one player's face-up card is higher than their opponent's.

If you want to have a go at playing it go ahead and try it here . In this problem, use the rules provided above.

Let's demonstrate with an example. You put down a queen from the top of your deck, while the opponent puts down a king.

You lose! The opponent gets both the cards.

Another example with pictures:

The player gets both cards!

Calculate the expected duration of the game (i.e. how many turns there will be). Then click on the range in which your answer falls.

Details and Assumptions:

• I do not know of a combinatorial method for solving this problem; hence I have tagged it under Computer Science. If anyone can find a combinatorial solution please post it so that the Brilliant community and I may have a look. For now, I think a combinatorial solution would be very arduous and strenuous to create.

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