Base 9: Summary: All Digit Sets

Blog: The First Pixel: Kaprekar's Constant 6174

Summary
[All Digit Sets] — [2] [3] [4] [5] [6] [7] [8] [Base 9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134] [135] [136] [137] [138] [139] [140] [141] [142] [143] [144] [145] [146] [147] [148] [149] [150] [151] [152] [153] [154] [155] [156] [157] [158] [159] [160] [161] [162] [163] [164] [165] [166] [167] [168] [169] [170] [171] [172] [173] [174] [175] [176] [177] [178] [179] [180] [181] [182] [183] [184] [185] [186] [187] [188] [189] [190] [191] [192] [193] [194] [195] [196] [197] [198] [199] [200] [201] [202] [203] [204] [205] [206] [207] [208] [209] [210] [211] [212] [213] [214] [215] [216] [217] [218] [219] [220] [221] [222] [223] [224] [225] [226] [227] [228] [229] [230] [231] [232] [233] [234] [235] [236] [237] [238] [239] [240] [241] [242] [243] [244] [245] [246] [247] [248] [249] [250] [251] [252] [253] [254] [255]
[Odd Digit Sets]
[Even Digit Sets]
Final Numbers
[Fixed Points]
Cycles Lengths
[All Digit Sets]
[Odd Digit Sets]
[Even Digit Sets]
Iteration Frequencies
[Cycle Start]
[Full Cycle]
Maximum Iteration Sequences
[Cycle Start]
[Full Cycle]
Cycles Sequences
[Maximum Length Cycle Count]
[All Cycle Counts]

Radix Character Encoding
0	1	2	3	4	5	6	7	8

Color Key
Orange	Fixed Point	number that becomes itself after one iteration
Green	Final Numbers	all numbers eventually reached, from all cycles
Purple	Cycle Start	iterations include entry into cycle, assuming it repeats
Red	Full Cycle	iterations include entire cycle, proving it repeats
Blue	Start Numbers	subset (optimization) of all numbers; minimal for full coverage
Gray	No Results	any calculation that has no results

Digits	# Start Numbers	# Fixed Points (excluding zero)	# Cycles (including fixed) (excluding zero)	Full Cycle Max Length	Avg Steps including Cycle Start	Max Steps including Cycle Start	Avg Steps including Full Cycle	Max Steps including Full Cycle	# Final Numbers (excluding zero)
1	9	0 fixed pts	0 cycles	0 nodes	0.89 avg steps	1 max steps	1.89 avg steps	2 max steps	0 final numbers
2	45	0 fixed pts	1 cycle	2 nodes	1.89 avg steps	3 max steps	3.60 avg steps	5 max steps	2 final numbers
3	165	0 fixed pts	1 cycle	2 nodes	2.54 avg steps	5 max steps	4.48 avg steps	7 max steps	2 final numbers
4	495	0 fixed pts	2 cycles	3 nodes	2.18 avg steps	4 max steps	5.15 avg steps	7 max steps	6 final numbers
5	1,287	1 fixed pt	2 cycles	5 nodes	3.00 avg steps	7 max steps	7.68 avg steps	12 max steps	6 final numbers
6	3,003	0 fixed pts	1 cycle	14 nodes	3.42 avg steps	8 max steps	17.39 avg steps	22 max steps	14 final numbers
7	6,435	0 fixed pts	1 cycle	2 nodes	6.86 avg steps	11 max steps	8.86 avg steps	13 max steps	2 final numbers
8	12,870	1 fixed pt	2 cycles	4 nodes	10.96 avg steps	25 max steps	14.91 avg steps	29 max steps	5 final numbers
9	24,310	1 fixed pt	2 cycles	12 nodes	5.06 avg steps	11 max steps	15.69 avg steps	23 max steps	13 final numbers
10	43,758	1 fixed pt	3 cycles	5 nodes	7.55 avg steps	21 max steps	11.68 avg steps	25 max steps	10 final numbers
11	75,582	1 fixed pt	2 cycles	6 nodes	8.96 avg steps	21 max steps	14.94 avg steps	27 max steps	7 final numbers
12	125,970	0 fixed pts	3 cycles	6 nodes	7.93 avg steps	18 max steps	10.35 avg steps	20 max steps	10 final numbers
13	203,490	1 fixed pt	3 cycles	6 nodes	7.27 avg steps	18 max steps	11.54 avg steps	22 max steps	11 final numbers
14	319,770	2 fixed pts	3 cycles	11 nodes	9.68 avg steps	31 max steps	20.54 avg steps	42 max steps	13 final numbers
15	490,314	1 fixed pt	4 cycles	6 nodes	6.59 avg steps	19 max steps	10.83 avg steps	23 max steps	14 final numbers
16	735,471	1 fixed pt	2 cycles	2 nodes	11.79 avg steps	27 max steps	13.79 avg steps	29 max steps	3 final numbers
17	1,081,575	2 fixed pts	4 cycles	7 nodes	8.02 avg steps	26 max steps	12.72 avg steps	27 max steps	11 final numbers
18	1,562,275	2 fixed pts	4 cycles	9 nodes	16.94 avg steps	52 max steps	25.16 avg steps	61 max steps	17 final numbers
19	2,220,075	3 fixed pts	5 cycles	4 nodes	8.70 avg steps	27 max steps	12.62 avg steps	31 max steps	11 final numbers
20	3,108,105	2 fixed pts	4 cycles	15 nodes	9.96 avg steps	30 max steps	24.89 avg steps	45 max steps	23 final numbers
21	4,292,145	2 fixed pts	6 cycles	7 nodes	7.21 avg steps	24 max steps	13.42 avg steps	30 max steps	23 final numbers
22	5,852,925	1 fixed pt	5 cycles	10 nodes	21.98 avg steps	52 max steps	31.72 avg steps	62 max steps	17 final numbers
23	7,888,725	4 fixed pts	9 cycles	8 nodes	10.64 avg steps	30 max steps	18.01 avg steps	38 max steps	28 final numbers
24	10,518,300	2 fixed pts	4 cycles	6 nodes	18.93 avg steps	45 max steps	21.62 avg steps	47 max steps	10 final numbers
25	13,884,156	3 fixed pts	15 cycles	8 nodes	7.69 avg steps	27 max steps	12.01 avg steps	32 max steps	58 final numbers
26	18,156,204	2 fixed pts	4 cycles	13 nodes	17.99 avg steps	50 max steps	25.92 avg steps	61 max steps	17 final numbers
27	23,535,820	3 fixed pts	15 cycles	20 nodes	8.94 avg steps	34 max steps	24.78 avg steps	54 max steps	69 final numbers
28	30,260,340	3 fixed pts	5 cycles	31 nodes	14.05 avg steps	45 max steps	44.20 avg steps	76 max steps	40 final numbers
29	38,608,020	4 fixed pts	18 cycles	8 nodes	14.38 avg steps	47 max steps	21.72 avg steps	55 max steps	71 final numbers
30	48,903,492	2 fixed pts	5 cycles	22 nodes	14.27 avg steps	59 max steps	30.55 avg steps	81 max steps	38 final numbers
31	61,523,748	3 fixed pts	19 cycles	8 nodes	13.05 avg steps	38 max steps	20.08 avg steps	46 max steps	83 final numbers
32	76,904,685	4 fixed pts	7 cycles	35 nodes	15.07 avg steps	49 max steps	40.85 avg steps	84 max steps	43 final numbers
33	95,548,245	4 fixed pts	23 cycles	29 nodes	10.17 avg steps	32 max steps	31.42 avg steps	60 max steps	125 final numbers
34	118,030,185	4 fixed pts	5 cycles	22 nodes	25.50 avg steps	65 max steps	47.36 avg steps	87 max steps	26 final numbers
35	145,008,513	4 fixed pts	30 cycles	8 nodes	17.37 avg steps	65 max steps	22.95 avg steps	70 max steps	139 final numbers
36	177,232,627	4 fixed pts	6 cycles	9 nodes	19.90 avg steps	57 max steps	27.38 avg steps	66 max steps	17 final numbers
37	215,553,195	4 fixed pts	32 cycles	18 nodes	21.52 avg steps	62 max steps	35.89 avg steps	80 max steps	154 final numbers
38	260,932,815	5 fixed pts	8 cycles	7 nodes	20.34 avg steps	62 max steps	27.23 avg steps	69 max steps	23 final numbers
39	314,457,495	4 fixed pts	32 cycles	18 nodes	21.21 avg steps	64 max steps	35.36 avg steps	82 max steps	167 final numbers
40	377,348,994	4 fixed pts	8 cycles	21 nodes	14.20 avg steps	48 max steps	24.93 avg steps	60 max steps	42 final numbers
41	450,978,066	5 fixed pts	34 cycles	39 nodes	20.71 avg steps	70 max steps	54.38 avg steps	109 max steps	202 final numbers
42	536,878,650	4 fixed pts	11 cycles	22 nodes	14.60 avg steps	46 max steps	27.12 avg steps	62 max steps	71 final numbers
43	636,763,050	5 fixed pts	34 cycles	27 nodes	26.95 avg steps	98 max steps	51.74 avg steps	125 max steps	204 final numbers
44	752,538,150	5 fixed pts	11 cycles	8 nodes	22.32 avg steps	65 max steps	29.81 avg steps	73 max steps	34 final numbers
45	886,322,710	5 fixed pts	39 cycles	16 nodes	22.29 avg steps	94 max steps	30.57 avg steps	101 max steps	209 final numbers
46	1,040,465,790	5 fixed pts	12 cycles	22 nodes	16.72 avg steps	81 max steps	35.86 avg steps	103 max steps	65 final numbers
47	1,217,566,350	6 fixed pts	42 cycles	31 nodes	26.01 avg steps	108 max steps	44.32 avg steps	139 max steps	227 final numbers
48	1,420,494,075	5 fixed pts	14 cycles	8 nodes	26.51 avg steps	89 max steps	29.62 avg steps	91 max steps	45 final numbers
49	1,652,411,475	6 fixed pts	43 cycles	15 nodes	34.18 avg steps	153 max steps	45.17 avg steps	166 max steps	212 final numbers
50	1,916,797,311	6 fixed pts	15 cycles	21 nodes	22.83 avg steps	87 max steps	42.28 avg steps	108 max steps	62 final numbers
51	2,217,471,399	6 fixed pts	44 cycles	15 nodes	50.60 avg steps	208 max steps	59.97 avg steps	218 max steps	223 final numbers
52	2,558,620,845	5 fixed pts	17 cycles	35 nodes	24.10 avg steps	81 max steps	40.65 avg steps	97 max steps	86 final numbers
53	2,944,827,765	7 fixed pts	50 cycles	15 nodes	56.28 avg steps	238 max steps	61.89 avg steps	243 max steps	243 final numbers
54	3,381,098,545	6 fixed pts	19 cycles	13 nodes	16.77 avg steps	65 max steps	25.91 avg steps	78 max steps	77 final numbers
55	3,872,894,697	7 fixed pts	54 cycles	65 nodes	45.06 avg steps	207 max steps	94.88 avg steps	272 max steps	317 final numbers
56	4,426,165,368	6 fixed pts	22 cycles	10 nodes	21.76 avg steps	72 max steps	28.67 avg steps	80 max steps	90 final numbers
57	5,047,381,560	6 fixed pts	57 cycles	28 nodes	41.88 avg steps	236 max steps	62.22 avg steps	264 max steps	310 final numbers
58	5,743,572,120	6 fixed pts	22 cycles	40 nodes	15.34 avg steps	62 max steps	47.45 avg steps	102 max steps	125 final numbers
59	6,522,361,560	8 fixed pts	63 cycles	21 nodes	67.93 avg steps	315 max steps	77.40 avg steps	324 max steps	324 final numbers
60	7,392,009,768	6 fixed pts	21 cycles	8 nodes	25.17 avg steps	76 max steps	32.81 avg steps	84 max steps	83 final numbers
61	8,361,453,672	7 fixed pts	64 cycles	21 nodes	88.86 avg steps	362 max steps	106.56 avg steps	382 max steps	363 final numbers
62	9,440,350,920	7 fixed pts	23 cycles	8 nodes	38.84 avg steps	108 max steps	40.54 avg steps	109 max steps	86 final numbers
63	10,639,125,640	8 fixed pts	68 cycles	21 nodes	56.15 avg steps	305 max steps	61.43 avg steps	309 max steps	346 final numbers
64	11,969,016,345	8 fixed pts	29 cycles	46 nodes	26.11 avg steps	96 max steps	54.19 avg steps	142 max steps	159 final numbers
65	13,442,126,049	8 fixed pts	95 cycles	21 nodes	42.40 avg steps	305 max steps	51.14 avg steps	309 max steps	492 final numbers
66	15,071,474,661	7 fixed pts	26 cycles	25 nodes	36.23 avg steps	111 max steps	59.14 avg steps	136 max steps	124 final numbers
67	16,871,053,725	7 fixed pts	127 cycles	21 nodes	39.50 avg steps	305 max steps	45.68 avg steps	309 max steps	676 final numbers
68	18,855,883,575	9 fixed pts	34 cycles	21 nodes	24.47 avg steps	116 max steps	34.38 avg steps	121 max steps	152 final numbers
69	21,042,072,975	8 fixed pts	130 cycles	23 nodes	46.44 avg steps	305 max steps	59.05 avg steps	309 max steps	706 final numbers
70	23,446,881,315	8 fixed pts	34 cycles	8 nodes	38.14 avg steps	125 max steps	43.67 avg steps	133 max steps	142 final numbers
71	26,088,783,435	8 fixed pts	136 cycles	37 nodes	46.66 avg steps	305 max steps	69.32 avg steps	309 max steps	780 final numbers
72	28,987,537,150	8 fixed pts	36 cycles	37 nodes	34.16 avg steps	117 max steps	65.75 avg steps	154 max steps	177 final numbers
73	32,164,253,550	9 fixed pts	140 cycles	32 nodes	45.19 avg steps	305 max steps	61.62 avg steps	309 max steps	794 final numbers
74	35,641,470,150	9 fixed pts	39 cycles	10 nodes	24.44 avg steps	108 max steps	31.59 avg steps	115 max steps	172 final numbers
75	39,443,226,966	8 fixed pts	142 cycles	23 nodes	85.58 avg steps	305 max steps	95.82 avg steps	309 max steps	798 final numbers
76	43,595,145,594	8 fixed pts	37 cycles	15 nodes	43.45 avg steps	173 max steps	56.81 avg steps	188 max steps	162 final numbers
77	48,124,511,370	10 fixed pts	151 cycles	36 nodes	46.33 avg steps	305 max steps	69.42 avg steps	309 max steps	903 final numbers
78	53,060,358,690	9 fixed pts	38 cycles	15 nodes	30.69 avg steps	121 max steps	37.73 avg steps	128 max steps	165 final numbers