# Fun With Numbers

### by Corey Plover

ANSWER: **RUN UP A SCORE** / **FEEL LIKE A MILLION**

Zap Kaplow's final levels attained on each game are listed in the table below.

The supplementary information from 1983 reveals that in the alternate history, aliens reprogrammed all Atari games to use a base 27 numbering system, therefore changing the final levels attained on many of the games.

Game | Description | Base 10 | Base 27 |
---|---|---|---|

Asteroids | number of 2 digit, happy numbers | 17 | (21)_{10} = L |

Berserk | sum of the digits in the largest 3 digit, palindromic, nonagonal number | 24 | (58)_{10} = 24 |

Centipede | sum of the digits in the largest 3 digit, palindromic, centered decagonal number | 7 | (35)_{10} = 18 |

Defender | number of up-and/or-right path from lower left to upper right of a 5x5 chessboard | 321 | (321)_{10} = BO |

Enduro | number of 2 digit, happy emirps | 4 | (4)_{10} = 4 |

Frogger | 6th Bell number | 203 | (203)_{10} = 7E |

Galaxian | number of straight lines on a smooth cubic surface | 27 | (27)_{10} = 10 |

Hangman | smallest highly cototient, Carol number | 47 | (47)_{10} = 1K |

Indy 500 | number of 3 digit, equidigital, heptagonal numbers | 4 | (7)_{10} = 7 |

Joust | only non-trivial Fibonacci number that is a perfect cube | 8 | (8)_{10} = 8 |

Keystone Kapers | number of 3 digit, emirps | 28 | (544)_{10} = K4 |

Lock 'N Chase | only positive integer lying directly between a square and a cube | 26 | (26)_{10} = Q |

Missile Command | smallest number whose first 3 multiples all contain the digit 3 | 1153 | (825)_{10} = 13F |

No Escape! | number of 2 digit, self, squares | 1 | (10)_{10} = A |

Off the Wall | smallest Mersenne prime counterexample | 11 | (11)_{10} = B |

Pacman | smallest 2 digit, palindromic, Kaprekar number | 55 | (560)_{10} = KK |

Q*Bert | smallest 2 digit, palindromic, Heptagonal number | 55 | (112)_{10} = 44 |

Road Runner | number of orderings of 4 objects with ties allowed | 75 | (75)_{10} = 2L |

Space Invaders | smallest number n so that the largest prime factors of n through n + 4 decrease | 491 | (491)_{10} = I5 |

Tempest | number of unique arrangements of 8 non-attacking Queens on an 8x8 chessboard | 92 | (92)_{10} = 3B |

Up 'N Down | number of 4-digit true vampire numbers | 7 | (9)_{10} = 9 |

Vanguard | only prime triangular number | 3 | (3)_{10} = 3 |

Wizard of Wor | smallest number expressible as the sum of 4 non-zero squares in exactly 9 ways | 162 | (162)_{10} = 60 |

Xenophobe | sum of all 2 digit, equidigital, tetrahedral numbers | 45 | (35)_{10} = 18 |

Yars' Revenge | number of 2 digit, happy, odious numbers | 13 | (12)_{10} = C |

Z-Tack | number of Archimedian solids (excluding the two chiral forms) | 13 | (13)_{10} = D |

Inserting these values into the formulas gives value 182114, 2116, 1 and 1903151805 in the original history (which translates to **RUN UP A SCORE** by taking pairs of numbers alphanumerically) and **FEEL LIKE A MILLION** directly in the alternate history (using 0-9A-Q notation).

Notes:

- Berserk is the digit sum of 969 for base 10 and (13734)
_{10}= IMI for base 27 - Centipede is the digit sum of 151 for base 10 and (11761)
_{10}= G3G for base 27 - Enduro comprises 13, 31, 79 and 97 for base 10 and 7M, DM, M7 and MD for base 27
- Indy 500 comprises 112, 189, 235 and 469 for base 10 and (874)
_{10}= 15A, (1177)_{10}= 1GG, (1651)_{10}= 274, (2673)_{10}= 3IO, (2839)_{10}= 3O4, (3553)_{10}= 4NG and (5221)_{10}= 74A for base 27 - Missile Command is 1153 (multiples are 2306, 3459) for base 10 and (825)
_{10}= 13F (multiples are (1650)_{10}= 273 and (2475)_{10}= 3AI) for base 27 - No Escape! comprises 64 for base 10 and (49)
_{10}= 1M, (81)_{10}= 30=, (121)_{10}= 4D, (169)_{10}= 67, (225)_{10}= 89, (289)_{10}= AJ, (361)_{10}= DA, (441)_{10}= G9, (529)_{10}= JG and (625)_{10}= N4 for base 27 - Up 'N Down comprises 1260 = 21 x 60, 1395 = 15 x 93, 1435 = 41 x 53, 1530 = 51 x 30, 1827 = 21 x 87, 2187 = 27 x 81 and 6880 = 80 x 86 for base 10 and 12BI = 2I x B1, 13A0 = 31 x A0, 147C = 4C x 71, 165G = 51 x 6G, 1940 = 40 x 91, 1AOF = 1F x OA, 1C3O = 31 x CO, 2DPI = 2I x PD and 539F = 93 x F5 for base 27
- Wizard of Wor is 162 = sums of squares of {12,4,1,1}, {11,6,2,1}, {11,4,4,3}, {10,7,3,2}, {10,6,5,1}, {9,8,4,1}, {9,7,4,4}, {9,6,6,3} and {8,8,5,3}
- Xenophobe comprises 10 and 35 for base 10 and (35)
_{10}= 18 for base 27 - Yar's Revenge comprises 13, 19, 28, 31, 32, 44, 49, 70, 79, 82, 91, 94 and 97 for base 10 and 3M, 7M, CF, DI, I7, ID, M3, M7, MD, N2, O5 and OH for base 27