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.

GameDescriptionBase 10Base 27
Asteroidsnumber of 2 digit, happy numbers17(21)10 = L
Berserksum of the digits in the largest 3 digit, palindromic, nonagonal number24(58)10 = 24
Centipedesum of the digits in the largest 3 digit, palindromic, centered decagonal number7(35)10 = 18
Defendernumber of up-and/or-right path from lower left to upper right of a 5x5 chessboard321(321)10 = BO
Enduronumber of 2 digit, happy emirps4(4)10 = 4
Frogger6th Bell number203(203)10 = 7E
Galaxiannumber of straight lines on a smooth cubic surface27(27)10 = 10
Hangmansmallest highly cototient, Carol number47(47)10 = 1K
Indy 500number of 3 digit, equidigital, heptagonal numbers4(7)10 = 7
Joustonly non-trivial Fibonacci number that is a perfect cube8(8)10 = 8
Keystone Kapersnumber of 3 digit, emirps28(544)10 = K4
Lock 'N Chaseonly positive integer lying directly between a square and a cube26(26)10 = Q
Missile Commandsmallest number whose first 3 multiples all contain the digit 31153(825)10 = 13F
No Escape!number of 2 digit, self, squares1(10)10 = A
Off the Wallsmallest Mersenne prime counterexample11(11)10 = B
Pacmansmallest 2 digit, palindromic, Kaprekar number55(560)10 = KK
Q*Bertsmallest 2 digit, palindromic, Heptagonal number55(112)10 = 44
Road Runnernumber of orderings of 4 objects with ties allowed75(75)10 = 2L
Space Invaderssmallest number n so that the largest prime factors of n through n + 4 decrease491(491)10 = I5
Tempestnumber of unique arrangements of 8 non-attacking Queens on an 8x8 chessboard92(92)10 = 3B
Up 'N Downnumber of 4-digit true vampire numbers7(9)10 = 9
Vanguardonly prime triangular number3(3)10 = 3
Wizard of Worsmallest number expressible as the sum of 4 non-zero squares in exactly 9 ways162(162)10 = 60
Xenophobesum of all 2 digit, equidigital, tetrahedral numbers45(35)10 = 18
Yars' Revengenumber of 2 digit, happy, odious numbers13(12)10 = C
Z-Tacknumber 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