On Numbers and Games: Solution

By Greg Pliska and Guy Jacobson

Answer: MIDDLE KINGDOM

The title of the puzzle is the name of a well-known book by mathematician John Horton Conway, in which he discusses (among other things) combinatorial games, the Sprague-Grundy theorem, and in particular the game of Nim and nimbers. The game of Nim is played with “heaps” of items, with the number of items in each heap being important. Nim is not itself directly involved with this puzzle, although concepts from that game will be needed to solve.

Each of the Emperor’s misguided reviews (styled as posts on the Board Game Geek web
site) describes a popular game—or at least presents what the Emperor
*thinks* the game is about. The reviews are always posted in the
year of the game’s release, and they are listed
in alphabetical order of the game titles.

In order of appearance, the games reviewed are: Agricola, Bruges, Dice Town, Genoa, Go, Imperial, Istanbul, Lost Cities, Love Letter, Medici, Morels, Notre Dame, Saboteur, Set, and Uno. Each of these review posts recieved a unique number of thumbs from 1 through 15; ordering the game titles by thumb count and reading the first letters spells NIMSUMALLBGGIDS, which is meant to be parsed as the instruction “Nim-sum all BGG IDs.”

Nim-sum is a concept that is important in finding a winning strategy for the game of Nim (or any impartial game). The whole idea of nim-summing involves combining multiple impartial games into a single game, and associating an integer with [a position in] the game.

On Board Game Geek (aka BGG), every game gets a unique ID number, which is readily seen in the URL of the page associated with that game. For the set of games reviewed:

BGG ID | Game | Thumbs | Year |
---|---|---|---|

31260 | Agricola | 7 | 2007 |

136888 | Bruges | 10 | 2013 |

40793 | Dice Town | 14 | 2009 |

1345 | Genoa | 11 | 2001 |

188 | Go | 12 | -2200 |

24181 | Imperial | 2 | 2006 |

148949 | Istanbul | 13 | 2014 |

50 | Lost Cities | 8 | 1999 |

129622 | Love Letter | 9 | 2012 |

46 | Medici | 3 | 1995 |

122298 | Morels | 6 | 2012 |

25554 | Notre Dame | 1 | 2007 |

9220 | Saboteur | 15 | 2004 |

1198 | Set | 4 | 1988 |

2223 | Uno | 5 | 1971 |

To nim-sum of a set of numbers, we simply take the the bitwise XOR (⊕)
of their binary representations. Taking the BGG ID numbers of the games in the
reviews and nim-summing them (that is, XORing them all
together) forms a number that is the BGG ID of another game.

So 31260
⊕
136888
⊕
40793
⊕
1345
⊕
188
⊕
24181
⊕
148949
⊕
50
⊕
129622
⊕
46
⊕
122298
⊕
25554
⊕
9220
⊕
1198
⊕
2223
=
**33159**, which is the BGG ID number of **MIDDLE KINGDOM**, the answer to the puzzle.