Several Simple Pair Removal Games

December 3, 2014

The games differ only in their removal rules and in the number of cards dealt to the table:

Game Cards

DealtPairing Rule Simple Pairs (Rules) 9 Pairs of the same rank Nines (Rules) 9 Pairs that add to 9, singleton 9s, or JQK Tens (Rules) 13 Pairs that add to 10, four tens, four Jacks, four Queens, or four Kings Block Ten (Rules) 9 Pairs that add to 10, two Jacks, two Queens, or two Kings (tens cannot be removed) Elevens (Rules) 9 Pairs that add to 11, or JQK Suit Elevens (Rules) 15 Pairs of the same suit that add to 11, or JQK of the same suit Thirteens (Rules) 10 Pairs that add to 13, or singleton kings

Note that in "Block Ten" you win if you get all but four cards off, since the tens are not removable.

The game Fifteens could be classified with these, but there is actually a modicum of strategy to that, since cards can be paired in different ways, so it's analysis is a bit more complex.

Game | Percent Winnable | Average Cards Removed | Percent DOA |
---|---|---|---|

Simple Pairs | 24.3% | 21.6 | 5.10% |

Nines | 5.56% | 11.1 | 5.55% |

Tens | 9.13% | 14.7 | 2.26% |

Block Ten | 3.67% | 12.0 | 6.65% |

Elevens | 10.7% | 13.8 | 8.25% |

Suit Elevens | 13.1% | 12.1 | 11.2% |

Thirteens | 62.8% | 37.3 | 0.57% |

Obviously two factors are working here. Some pairing rules are more difficult than others, but some games have more tableau slots, making them easier.

To separate those two factors, I ran the solvers for each game using a range of different numbers of tableau slots. A million runs were done for each case. The percent winnable is reported for each game in the table below. The normal number of tableau slots for each game is highlighted in yellow. When "0%" or "100%" is in bold, then there were literally no games won or lost in the million game sample.

Pairing Rule | Number of Tableau Slots | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | |

Suit Elevens | 0.00% |
0.00% |
0.00% |
0.00% |
0.00% | 0.01% | 0.09% | 0.46% | 1.84% | 5.52% | 13.14% | 25.51% | 41.75% | 59.23% | 75.00% | 86.66% |

Tens | 0.00% |
0.00% |
0.00% |
0.00% | 0.01% | 0.13% | 0.79% | 3.12% | 9.13% | 20.40% | 36.67% | 55.37% | 72.75% | 85.77% | 93.71% | 97.65% |

Nines | 0.00% |
0.00% | 0.08% | 1.04% | 5.56% | 16.57% | 33.76% | 53.12% | 70.51% | 83.43% | 91.61% | 96.21% | 98.48% | 99.45% | 99.83% | 99.95% |

Block Ten | 0.00% |
0.00% | 0.03% | 0.53% | 3.67% | 13.77% | 32.84% | 56.24% | 76.41% | 89.43% | 96.07% | 98.78% | 99.69% | 99.94% | 99.99% | 100.00% |

Elevens | 0.00% |
0.01% | 0.28% | 2.63% | 10.74% | 26.57% | 46.56% | 66.60% | 81.54% | 91.02% | 96.17% | 98.55% | 99.52% | 99.85% | 99.96% | 99.99% |

Simple Pairs | 0.00% | 0.01% | 0.49% | 5.43% | 24.32% | 56.72% | 84.33% | 96.76% | 99.71% | 100.00% |
100.00% |
100.00% |
100.00% |
100.00% |
100.00% |
100.00% |

Thirteens | 0.01% | 0.52% | 4.72% | 18.00% | 39.76% | 62.82% | 80.64% | 91.44% | 96.81% | 98.98% | 99.72% | 99.93% | 99.99% | 100.00% | 100.00% | 100.00% |

The "Thirteens" pairing rules seems to be easiest (at least while the number of slots is small), and it also seems to be the most widely used, being used in games such as Pyramid, Links, and Friday.

Note that in "Simple Pairs" there are three different possible matches for any given card (the other three cards of the same suit), while in "Thirteens" there are four possible matches (the four card that add to 13 with it) for most cards, and the kings don't need a match. So it's not surprising that that "Thirteens" is easier.

But as the number of slots gets large, "Simple Pairs" gets easier. With "Simple Pairs" it is impossible to have 14 cards without a pair, so naturally the win rate is 100% if you have more than 13 slots. But in "Thirteens" you could have as many as 24 cards without a match (all the aces, twos, threes, fours, fives, and sixes), so though each card has more matches, it is possible to have larger sets with no matches.