# The Math of Patches

Hey everyone,

Here today with something a little bit different, an article on the card “Patches” from the popular card game Hearthstone. Very often, quantitative analysis of cards is difficult to achieve because of the diverse variety of decks that a single card can be played in, as well as the situation surrounding when the card is played.

“Patches” is a card that is activated when a card from a “pirate” is played. A “pirate” is a subset of the card pool that we’ll refer to as an activator for purpose of general use. Patches is removed from the deck, and pulled directly into play if a pirate is played. While the benefit of this cannot be understated, there is a worst-case scenario:

If Patches is in the hand when an activator is played, the card is NOT removed from the deck (as it is in your hand) and you essentially lose a drawn card that you otherwise could’ve had, if Patches was played for free. Thus, calculation for the worst-case scenario is incredibly important. In the paper, I’ve included the number of activators vs. the chances of not drawing Patches so the player can evaluate the optimal amount of activators to put into the deck.

### Introductory Information for the Game:

When the game begins, players are randomized where one player is picked to go first and the other to go second. Like chess, playing first has the advantage of “tempo”, giving a slight edge to the player who has the option to play a card first. Thus, the second player receives an extra card in the “mulligan phase” to compensate. During the “mulligan phase”, players can decide what cards to keep and what cards they wish to discard. Cards discarded are placed in a distinct group from the rest of the deck for the rest of the mulligan, but returned into the deck after the mulligan. The number of cards mulliganned is then redrawn from the reduced deck without the cards that were mulliganned. Cards mulliganned cannot be redrawn into the hand. This becomes more important in calculation of the probability of drawing Patches later on.

These are a few examples of salient calculations of probability available in the paper. I look forward to hearing your thoughts!

Link to the PDF: The Math of Patches

Happy Hearthstoning!

Joseph

## One Reply to “The Math of Patches”

1. Reblogged this on SMART and commented:
An insightful post from my friend, Joseph! Check out his blog and happy hearthstoning – maybe figure out the math while you’re at it for an extra fun challenge 🙂

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