Delete and Repeat till Complete

by Simon Whitechapel

265 + 652 = 827; 827 + 728 = 1555; 1555 + 5551 = 7106; 7106 + 6017 = 13123; 13123 + 32131 = 45254; 45254 + 45254 = 90508; 90508 + 80509 = 171017...

Strictly speaking, the sequence never ends: the numbers just get larger and larger. Mathematically speaking, however, something interesting has happened at the fifth step, 13123 + 32131 = 45254. 45254 is interesting because it's a palindrome: it reads the same forwards as backwards. 265 isn't unusual: most small numbers quickly produce a palindrome and the number 196 is famous precisely because, so far, it hasn't, although the procedure has been carried to many thousands of steps, producing numbers many thousands of digits long.

And plainly, every number produced so far in that sequence has not ended in a palindrome either. Perhaps the sequence will carry on for ever without producing a palindrome. As yet no-one knows and no-one can prove the result one way or another, although it has been proved that a palindrome is never produced when the procedure is carried out in base two on the number 10110 (22 in base ten).¹

Much easier is a proof of the result when the procedure is carried out in reverse: think of a number; reverse it; subtract the higher from the lower; then repeat. There are only two possible outcomes: the sequence will end in zero, after a single-digit number or palindrome has been created; or the sequence will fall into an endless loop. Here are some examples of the first possibility:

10: 10 - 01 = 9; 9 - 9 = 0
12: 21 - 12 = 9; 9 - 9 = 0
13: 31 - 13 = 18; 81 - 18 = 63; 63 - 36 = 27; 72 - 27 = 45; 54 - 45 = 9; 9 - 9 = 0
14>27>45>9>0
15>36>27>45>9>0
16>45>9>0
17>54>9>0
68>18>63>27>45>9>0
70>63>27>45>9>0
75>18>63>27>45>9>0
743>396>297>495>99>0
745>198>693>297>495>99>0
5333>1998>6993>2997>4995>999>0
5337>1998>6993>2997>4995>999>0
12334>30987>47916>14058>70983>32076>34947>
   >39996>29997>49995>9999>0
12335>40986>27918>54054>9009>0
12342>11979>85932>61974>14058>70983>32076
   >34947>39996>29997>49995>9999>0

And here are some examples of the second possibility, in which the numbers fall into a loop:

1012>1089>8712>6534>2178>6534
1023>2178>6534>2178
10012>10989>87912>65934>21978>65934
10023>21978>65934>21978

These are all numbers in base 10. In certain other bases, the loop can be much shorter:

1012₃>1012₃>1012₃...
21201₃>21201₃>21201₃...

To see why this happens, examine these equations:

2101₃ - 1012₃ = 1012₃ (64₁₀ - 32₁₀ = 32₁₀)
21201₃ - 10212₃ = 10212₃ (208₁₀ - 104₁₀ = 104₁₀)

If x-y = y, x = y+y; that is 2y = x. This introduces the concept of multiplicative palindromes:

Base 3

1012₃ x 2 = 2101₃ (32₁₀ x 2 = 64₁₀)
10212₃ x 2 = 21201₃ (104₁₀ x 2 = 208₁₀)
102212₃ x 2 = 212201₃ (320₁₀ x 2 = 640₁₀)
1022212₃ x 2 = 2122201₃ (968₁₀ x 2 = 1936₁₀)

Base 4

1023₄ x 3 = 3201₄ (75₁₀ x 3 = 225₁₀)
10323₄ x 3 = 32301₄ (315₁₀ x 3 = 945₁₀)
103323₄ x 3 = 323301₄ (1275₁₀ x 3 = 3825₁₀)
1033323₄ x 3 = 3233301₄ (5115₁₀ x 3 = 15345₁₀)

Base 5

143₅ x 2 = 341₅ (48₁₀ x 2 = 96₁₀)
1313₅ x 2 = 3131₅ (208₁₀ x 2 = 416₁₀)
1443₅ x 2 = 3441₅ (248₁₀ x 2 = 496₁₀)
13013₅ x 2 = 31031₅ (1008₁₀ x 2 = 2016₁₀)
14443₅ x 2 = 34441₅ (1248₁₀ x 2 = 2496₁₀)
130013₅ x 2 = 310031₅ (5008₁₀ x 2 = 10016₁₀)
131313₅ x 2 = 313131₅ (5208₁₀ x 2 = 10416₁₀)
143143₅ x 2 = 341341₅ (6048₁₀ x 2 = 12096₁₀)
144443₅ x 2 = 344441₅ (6248₁₀ x 2 = 12496₁₀)
1300013₅ x 2 = 3100031₅ (25008₁₀ x 2 = 50016₁₀)
1314313₅ x 2 = 3134131₅ (26208₁₀ x 2 = 52416₁₀)
1430143₅ x 2 = 3410341₅ (30048₁₀ x 2 = 60096₁₀)
1444443₅ x 2 = 3444441₅ (31248₁₀ x 2 = 62496₁₀)
1034₅ x 4 = 4301₅ (144₁₀ x 4 = 576₁₀)
10434₅ x 4 = 43401₅ (744₁₀ x 4 = 2976₁₀)
104434₅ x 4 = 434401₅ (3744₁₀ x 4 = 14976₁₀)
1044434₅ x 4 = 4344401₅ (18744₁₀ x 4 = 74976₁₀)

Base 6

2134₆ x 2 = 4312₆ (490₁₀ x 2 = 980₁₀)
21534₆ x 2 = 43512₆ (3010₁₀ x 2 = 6020₁₀)
215534₆ x 2 = 435512₆ (18130₁₀ x 2 = 36260₁₀)
2155534₆ x 2 = 4355512₆ (108850₁₀ x 2 = 217700₁₀)
1045₆ x 5 = 5401₆ (245₁₀ x 5 = 1225₁₀)
10545₆ x 5 = 54501₆ (1505₁₀ x 5 = 7525₁₀)
105545₆ x 5 = 545501₆ (9065₁₀ x 5 = 45325₁₀)
1055545₆ x 5 = 5455501₆ (54425₁₀ x 5 = 272125₁₀)

Base 7

165₇ x 3 = 561₇ (96₁₀ x 3 = 288₁₀)
1515₇ x 3 = 5151₇ (600₁₀ x 3 = 1800₁₀)
1665₇ x 3 = 5661₇ (684₁₀ x 3 = 2052₁₀)
15015₇ x 3 = 51051₇ (4128₁₀ x 3 = 12384₁₀)
16665₇ x 3 = 56661₇ (4800₁₀ x 3 = 14400₁₀)
150015₇ x 3 = 510051₇ (28824₁₀ x 3 = 86472₁₀)
151515₇ x 3 = 515151₇ (29412₁₀ x 3 = 88236₁₀)
165165₇ x 3 = 561561₇ (33024₁₀ x 3 = 99072₁₀)
166665₇ x 3 = 566661₇ (33612₁₀ x 3 = 100836₁₀)
1500015₇ x 3 = 5100051₇ (201696₁₀ x 3 = 605088₁₀)
1516515₇ x 3 = 5156151₇ (206400₁₀ x 3 = 619200₁₀)
1650165₇ x 3 = 5610561₇ (230592₁₀ x 3 = 691776₁₀)
1666665₇ x 3 = 5666661₇ (235296₁₀ x 3 = 705888₁₀)
1056₇ x 6 = 6501₇ (384₁₀ x 6 = 2304₁₀)
10656₇ x 6 = 65601₇ (2736₁₀ x 6 = 16416₁₀)
106656₇ x 6 = 656601₇ (19200₁₀ x 6 = 115200₁₀)
1066656₇ x 6 = 6566601₇ (134448₁₀ x 6 = 806688₁₀)

Base 8

25₈ x 2 = 52₈ (21₁₀ x 2 = 42₁₀)
275₈ x 2 = 572₈ (189₁₀ x 2 = 378₁₀)
2525₈ x 2 = 5252₈ (1365₁₀ x 2 = 2730₁₀)
2775₈ x 2 = 5772₈ (1533₁₀ x 2 = 3066₁₀)
25025₈ x 2 = 52052₈ (10773₁₀ x 2 = 21546₁₀)
27775₈ x 2 = 57772₈ (12285₁₀ x 2 = 24570₁₀)
250025₈ x 2 = 520052₈ (86037₁₀ x 2 = 172074₁₀)
252525₈ x 2 = 525252₈ (87381₁₀ x 2 = 174762₁₀)
275275₈ x 2 = 572572₈ (96957₁₀ x 2 = 193914₁₀)
277775₈ x 2 = 577772₈ (98301₁₀ x 2 = 196602₁₀)
2500025₈ x 2 = 5200052₈ (688149₁₀ x 2 = 1376298₁₀)
2527525₈ x 2 = 5257252₈ (700245₁₀ x 2 = 1400490₁₀)
2750275₈ x 2 = 5720572₈ (774333₁₀ x 2 = 1548666₁₀)
2777775₈ x 2 = 5777772₈ (786429₁₀ x 2 = 1572858₁₀)
2156₈ x 3 = 6512₈ (1134₁₀ x 3 = 3402₁₀)
21756₈ x 3 = 65712₈ (9198₁₀ x 3 = 27594₁₀)
217756₈ x 3 = 657712₈ (73710₁₀ x 3 = 221130₁₀)
2177756₈ x 3 = 6577712₈ (589806₁₀ x 3 = 1769418₁₀)
1015₈ x 5 = 5101₈ (525₁₀ x 5 = 2625₁₀)
11165₈ x 5 = 56111₈ (4725₁₀ x 5 = 23625₁₀)
102515₈ x 5 = 515201₈ (34125₁₀ x 5 = 170625₁₀)
112665₈ x 5 = 566211₈ (38325₁₀ x 5 = 191625₁₀)
1016015₈ x 5 = 5106101₈ (269325₁₀ x 5 = 1346625₁₀)
1127665₈ x 5 = 5667211₈ (307125₁₀ x 5 = 1535625₁₀)
1067₈ x 7 = 7601₈ (567₁₀ x 7 = 3969₁₀)
10767₈ x 7 = 76701₈ (4599₁₀ x 7 = 32193₁₀)
107767₈ x 7 = 767701₈ (36855₁₀ x 7 = 257985₁₀)
1077767₈ x 7 = 7677701₈ (294903₁₀ x 7 = 2064321₁₀)

Base 9

3256₉ x 2 = 6523₉ (2400₁₀ x 2 = 4800₁₀)
32856₉ x 2 = 65823₉ (21840₁₀ x 2 = 43680₁₀)
328856₉ x 2 = 658823₉ (196800₁₀ x 2 = 393600₁₀)
3288856₉ x 2 = 6588823₉ (1771440₁₀ x 2 = 3542880₁₀)
187₉ x 4 = 781₉ (160₁₀ x 4 = 640₁₀)
1717₉ x 4 = 7171₉ (1312₁₀ x 4 = 5248₁₀)
1887₉ x 4 = 7881₉ (1456₁₀ x 4 = 5824₁₀)
17017₉ x 4 = 71071₉ (11680₁₀ x 4 = 46720₁₀)
18887₉ x 4 = 78881₉ (13120₁₀ x 4 = 52480₁₀)
170017₉ x 4 = 710071₉ (104992₁₀ x 4 = 419968₁₀)
171717₉ x 4 = 717171₉ (106288₁₀ x 4 = 425152₁₀)
187187₉ x 4 = 781781₉ (116800₁₀ x 4 = 467200₁₀)
188887₉ x 4 = 788881₉ (118096₁₀ x 4 = 472384₁₀)
1700017₉ x 4 = 7100071₉ (944800₁₀ x 4 = 3779200₁₀)
1718717₉ x 4 = 7178171₉ (957760₁₀ x 4 = 3831040₁₀)
1870187₉ x 4 = 7810781₉ (1049920₁₀ x 4 = 4199680₁₀)
1888887₉ x 4 = 7888881₉ (1062880₁₀ x 4 = 4251520₁₀)
1078₉ x 8 = 8701₉ (800₁₀ x 8 = 6400₁₀)
10878₉ x 8 = 87801₉ (7280₁₀ x 8 = 58240₁₀)
108878₉ x 8 = 878801₉ (65600₁₀ x 8 = 524800₁₀)
1088878₉ x 8 = 8788801₉ (590480₁₀ x 8 = 4723840₁₀)

Base 10

2178₁₀ x 4 = 8712₁₀
21978₁₀ x 4 = 87912₁₀
219978₁₀ x 4 = 879912₁₀
2199978₁₀ x 4 = 8799912₁₀
1089₁₀ x 9 = 9801₁₀
10989₁₀ x 9 = 98901₁₀
109989₁₀ x 9 = 989901₁₀
1099989₁₀ x 9 = 9899901₁₀

NOTE

1. The Penguin Dictionary of Curious and Interesting Numbers, David Wells, Penguin, London, 1987, entry for "196", pg. 143