*n*and

*d*be the numerator and the denominator of a fraction respectively such that I can "cancel" a digit

*D*in both

*n*and in

*d*and have the result =

*n*/

*d*. Usually this doesn't work for obvious reasons. 14/48, cancelling the 4s does not leave us with 7/24 but instead with 1/8, which is why we don't cancel digits when we are trying to simplify fractions! Famously, decimal 16/64, cancelling the 6s, actually does work: it does equal 1/4 like it is "supposed to". This is called "anomalous cancellation."

There is a trivial condition, that is the cancellation of trailing zeros. This is discounted regarding what we're interested in.

This sequence ports nicely into the dozenal condition and it might entertain folks here recreationally. The decimal case 16/64 = 1/4 is famous meme-wise; in dozenal the meme would be 1b;/b6; = 1/6.

The Wolfram language (11.1) code I wrote follows.

Code: Select all

```
With[{b = 12},
Apply[Join,
Parallelize@
Table[Map[{#, m} &, #] &@
Select[Range[b + 1, m - 1],
Function[k,
Function[{r, w, n, d},
AnyTrue[
Flatten@
Map[Apply[Outer[Divide, #1, #2] &, #] &,
Transpose@
MapAt[# /. 0 -> Nothing &,
Map[Function[x,
Map[Map[FromDigits[#, b] &@ Delete[x, #] &,
Position[x, #]] &, Intersection @@ {n, d}]], {n,
d}], -1]], # == Divide @@ {k, m} &]] @@ {k/m, #,
First@ #, Last@ #} &@ Map[IntegerDigits[#, b] &, {k, m}] -
Boole[Mod[{k, m}, b] == {0, 0}]] ], {m, b, b^3}]]]
```

Code: Select all

```
Dozenal fractions with nontrivial anomalous cancellation
All figures are dozenal after this sentence.
The trivial condition is that where N and D are both congruent to 0 (mod 10).
n = index.
N = Numerators of fractions with nontrivial anomalous cancellation.
D = Denominators of fractions with nontrivial anomalous cancellation.
listed with multiplicity if multiple numerators are possible.
r = ratio N/D
n N D r
-----------------------
1 1b b6 1/6
2 2b b8 1/4
3 3b b9 1/3
4 5b ba 1/2
5 11 110 1/10
6 12 120 1/10
7 22 121 2/11
8 13 130 1/10
9 33 132 3/12
a 14 140 1/10
b 44 143 4/13
10 15 150 1/10
11 55 154 5/14
12 16 160 1/10
13 66 165 6/15
14 17 170 1/10
15 77 176 7/16
16 18 180 1/10
17 88 187 8/17
18 19 190 1/10
19 99 198 9/18
1a 1a 1a0 1/10
1b aa 1a9 a/19
20 1b 1b0 1/10
21 b6 1b0 1/2
22 b7 1b2 1/2
23 2b 1b4 1/8
24 b8 1b4 1/2
25 3b 1b6 1/6
26 b9 1b6 1/2
27 5b 1b8 1/4
28 ba 1b8 1/2
29 7b 1b9 1/3
2a bb 1ba 1/2
2b 101 202 1/2
30 102 204 1/2
31 103 206 1/2
32 104 208 1/2
33 105 20a 1/2
34 21 210 1/10
35 22 220 1/10
36 121 220 11/20
37 23 230 1/10
38 33 231 3/21
39 132 231 12/21
3a 24 240 1/10
3b 44 242 2/11
40 143 242 13/22
41 25 250 1/10
42 55 253 5/23
43 154 253 14/23
44 26 260 1/10
45 66 264 3/12
46 165 264 15/24
47 27 270 1/10
48 77 275 7/25
49 176 275 16/25
4a 28 280 1/10
4b 88 286 4/13
50 187 286 17/26
51 29 290 1/10
52 99 297 9/27
53 198 297 18/27
54 2a 2a0 1/10
55 aa 2a8 5/14
56 1a9 2a8 19/28
57 2b 2b0 1/10
58 b8 2b0 1/3
59 1b4 2b0 2/3
5a 3b 2b3 1/9
5b b9 2b3 1/3
60 1b6 2b3 2/3
61 5b 2b6 1/6
62 ba 2b6 1/3
63 1b8 2b6 2/3
64 8b 2b8 1/4
65 bb 2b9 1/3
66 1ba 2b9 2/3
67 15b 2ba 1/2
68 101 303 1/3
69 202 303 2/3
6a 102 306 1/3
6b 204 306 2/3
70 103 309 1/3
71 206 309 2/3
72 31 310 1/10
73 32 320 1/10
74 33 330 1/10
75 132 330 7/16
76 231 330 21/30
77 34 340 1/10
78 44 341 4/31
79 143 341 13/31
7a 242 341 22/31
7b 139 346 7/16
80 35 350 1/10
81 55 352 5/32
82 154 352 8/17
83 253 352 23/32
84 36 360 1/10
85 66 363 2/11
86 165 363 15/33
87 264 363 24/33
88 37 370 1/10
89 77 374 7/34
8a 176 374 9/18
8b 275 374 25/34
90 38 380 1/10
91 88 385 8/35
92 187 385 17/35
93 286 385 26/35
94 39 390 1/10
95 99 396 3/12
96 198 396 a/19
97 297 396 27/36
98 3a 3a0 1/10
99 aa 3a7 a/37
9a 1a9 3a7 19/37
9b 2a8 3a7 28/37
a0 3b 3b0 1/10
a1 b9 3b0 1/4
a2 1b6 3b0 1/2
a3 2b3 3b0 3/4
a4 1b7 3b2 1/2
a5 5b 3b4 1/8
a6 ba 3b4 1/4
a7 1b8 3b4 1/2
a8 2b6 3b4 3/4
a9 7b 3b6 1/6
aa 1b9 3b6 1/2
ab bb 3b8 1/4
b0 1ba 3b8 1/2
b1 2b9 3b8 3/4
b2 13b 3b9 1/3
b3 1bb 3ba 1/2
b4 201 402 1/2
b5 101 404 1/4
b6 202 404 1/2
b7 303 404 3/4
b8 203 406 1/2
b9 102 408 1/4
ba 204 408 1/2
bb 306 408 3/4
100 205 40a 1/2
```