How to Play Sprouts with Playing Cards
It's not much fun to play sprouts with playing cards, but it is possible.
Why would anyone want to do such a thing? Maybe it will be easier to analyze
the game or more straightforward to implement it on a computer this way.
Maybe you're fresh out of pens and paper.
To play a game with N initial spots, you'll need three identical
decks of playing cards
with at least 3N+1 distinct cards in each deck. This way each card will have exactly two identical "twins". (Actually, the cards
are incidental; integers will suffice. This document is really about how to generate legal moves and
update a position represented in Dan Hoey's boundary-list notation.) In the
rules below, we'll assume you've combined the three decks into one large deck.
You will also need a flat space for at least 2N+1 rows of cards. A large
wooden table will do nicely. Each row will contain zero or more stacks of
cards, and each stack will contain one or more cards.
To "cut" a stack of cards means the usual thing: remove one or more cards
from the top of the stack (preserving order) and put them under the
remaining cards in the stack.
To "play" a card, remove it from the deck and place it in the specified
Initial position and play
The initial position consists of N different cards, each in an isolated
stack, all in the same row. Players take turns moving. There are two kinds of moves, joining moves and dividing moves. On his turn, a player must execute either a legal joining move or a legal dividing move, but not both in the same turn. Under the normal play convention, the last player wins. Under the misere play convention, the last player loses.
A joining move combines two stacks in the same row. If any step of the move is not possible, the move is illegal.
- Choose two different nonempty stacks A and B in the same row. Optionally, cut stack A. Optionally, cut stack B.
- If A has more than one card, then play a twin of the top card to the bottom of the stack.
- If B has more than one card, then play a twin of the top card to the bottom of the stack.
- From the deck, remove two identical cards that are not already on the table. Place one on top of A, and one on top of B.
- Place A on top of B.
A dividing move divides a stack into two stacks, one of which is placed in a
new row. One or more stacks from the original row may be moved to the new row. If any step of the move is not possible, the move is illegal.
- Choose a nonempty stack. Optionally, cut it.
- Play a twin of the top card to the bottom of the stack.
- Remove one or more cards -- but not all of them -- from the top of the stack and place them in a new row.
- If the original stack or the new stack has more than one card, then play a twin of the the bottom card of the new stack to the top of the original stack.
- From the deck, remove two identical cards that are not already on the table. Place one on top of the new stack, and one on top of the original stack.
- Optionally, move one or more stacks -- other than the original stack — from the original row to the new row.
Consider the following game (in WGOSA notation): 2+ 1(3)2 1(4)2 1(5)4 2(6)3
Here's how that game would be played with cards. To represent the position in text, we will use the following conventions: A stack of
cards is written in order from top card to bottom, with a comma between each
card. A semicolon separates each stack from adjacent stacks in the same row.
A slash separates each row from its neighbors. (This is the notation used by
Dan Hoey in his paper on sprouts notation.)
(2) initial position:
(3) joining move:
(4) dividing move:
(5) dividing move:
(6) dividing move:
Analogy with graph theory
We have represented a plane graph using rows of stacks of cards.
|Playing card term||Graph theory term
|card||occurrence of vertex in boundary
|stack||left-hand walk of vertices on boundary
|row||face, i.e. set of boundaries
|table||plane graph, i.e. set of faces
Author: Josh Purinton
This document builds upon the sprouts notation system devised by Dan Hoey.
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