Some numbers are more equivalent than others

Question

^{[Second part and bounty challenge appended in May 2020]}

          ALL ANIMALS ARE EQUAL
   BUT SOME ANIMALS ARE MORE EQUAL THAN OTHERS

         ^{— from Animal Farm by George Orwell}

Original puzzle from March 2019:

A contrived simple equivalence rule A applies neatly to numbers 0 through 99. (Application of this equivalence to further numbers is, well, equivocal.) All equivalences headed by numbers 0 through 19 are listed below, accounting for almost all other eligible numbers as well, where ‘=’ means “is equivalent to.” (Each number is reflexively equivalent to itself.)

$\require{begingroup}\begingroup \def\b #1{{ \bf\phantom{39}\llap{#1} }} \def\no {{ \textsf{no others} }} \def\= {{ \tiny \raise.3ex{\: = ~~} }} \small\begin{array}{llll} \textsf{Equivalence rule A:} ~~~ & \b{0} \= \no && \b{10}\= \no \\ & \b{1} \= \no && \b{11}\= 29 \= 31 \= 49 \= 51 \= 69 \= 71 \= 89 \= 91 \\ & \b{2} \= \no && \b{12}\= 28 \= 32 \= 48 \= 52 \= 68 \= 72 \= 88 \= 92 \\ & \b{3} \= \no && \b{13}\= 27 \= 33 \= 47 \= 53 \= 67 \= 73 \= 87 \= 93 \\ & \b{4} \= \no && \b{14}\= 26 \= 34 \= 46 \= 54 \= 66 \= 74 \= 86 \= 94 \\ & \b{5} \= \no && \b{15}\= 25 \= 35 \= 45 \= 55 \= 65 \= 75 \= 85 \= 95 \\ & \b{6} \= \no && \b{16}\= 24 \= 36 \= 44 \= 56 \= 64 \= 76 \= 84 \= 96 \\ & \b{7} \= \no && \b{17}\= 23 \= 37 \= 43 \= 57 \= 63 \= 77 \= 83 \= 97 \\ & \b{8} \= \no && \b{18}\= 22 \= 38 \= 42 \= 58 \= 62 \= 78 \= 82 \= 98 \\ & \b{9} \= \no && \b{19}\= 21 \= 39 \= 41 \= 59 \= 61 \= 79 \= 81 \= 99 \\ & && \b{20}\= \underline{~~~~~~~~} \dots \bf \, ? \\[-.5ex] & &~~~~& \phantom{2}\vdots \end{array}$

1. What, if any, are the equivalences of 20 by rule A?
   Please use and explain the simplest possible rule (which can be described in 11 words or less and is not purely mathematical) that accounts for every equivalence from 0 to 99.

Note: The equivalences listed above and below are complete and not equivalent to any other numbers.

Related puzzle, added partly to serve as a hint in May 2020:

A contrived simple equivalence rule B applies neatly to numbers 1 through 3999. (Application of this equivalence to further numbers is again equivocal.) All equivalences headed by numbers 1 through 39 are listed below, accounting for many other eligible numbers as well, with some extra spacing to help distinguish groups of consecutive numbers.

$\small\begin{array}{ll} \textsf{Equivalence rule B:} ~~ &\b{ 1} \= 2 \= 3 \\[.1ex] &\b{ 4} \= 5 \= 6 \= 7 \= 8 \\[.3ex] &\b{ 9} \= 10 \= 11 \= 12 \= 13 \\[.5ex] &\b{14} \= 15 \= 16 \= 17 \= 18 ~~ \= ~~ 50 \= 51 \= 52 \= 53 \\[.7ex] &\b{19} \= 20 \= 21 \= 22 \= 23 \kern1.1em \= ~~ 100 \= 101 \= 102 \= 103 \\[.9ex] &\b{24} \= 25 \= 26 \= 27 \= 28 \kern1.9em \= ~~ 40 \= 41 \= 42 \= 43 \\ &\b{} \= 59 \= 60 \= 61 \= 62 \= 63 \kern0.6em \= ~~ 104 \= 105 \= 106 \= 107 \= 108 \\ &\b{} \= 500 \= 501 \= 502 \= 503 \\[.9ex] &\b{29} \= 30 \= 31 \= 32 \= 33 \kern3.8em \= ~~ 90 \= 91 \= 92 \= 93 \\ &\b{} \= 109 \= 110 \= 111 \= 112 \= 113 ~~ \= ~~ 1000 \= 1001 \= 1002 \= 1003 \\[.9ex] &\b{34} \= 35 \= 36 \= 37 \= 38 \!\:~~~ \= ~~\!\: 49 ~~ \= ~~ 69 \= 70 \= 71 \= 72 \= 73 \\ &\b{} \= 94 \= 95 \= 96 \= 97 \= 98 \!\:~~~~ \= ~~ 114 \= 115 \= 116 \= 117 \= 118 \\ &\b{} \= 150 \= 151 \= 152 \= 153 ~~ \= ~~ 509 \= 510 \= 511 \= 512 \= 513 \\ &\b{} \= 1004 \= 1005 \= 1006 \= 1007 \= 1008 \\[.8ex] &\b{39} \= ~~ 99 ~~ \= ~~ 119 \= 120 \= 121 \= 122 \= 123 ~~ \= ~~ 200 \= 201 \= 202 \= 203 \\ &\b{} \= 1009 \= 1010 \= 1011 \= 1012 \= 1013 \\[.6ex] &\b{44} \= \underline{~~~~~~~~} \dots \bf \, ? \\[-.5ex] & \phantom{4}\vdots \end{array}\endgroup$

2. What, if any, are the equivalences of 44 by rule B?

3. Which number /numbers is /are the  l e a s t  equivalent (equivalent to the fewest others) by rule B?

Bounty challenge toughie, not required for a  ✓ correct answer but no-computers nonetheless:

4. Which numbers are the  m o s t  equivalent (equivalent to the most others) by rule B?
   (Anything close deserves votes of approval. A reasoned attempt deserves a bounty amount relative to thoroughness and correctness.)

Answer 1

Answer:

20 = no others

Reason: (humn has told me that this is wrong but it's my favorite guess of mine)

Because you gave us a list of equivalences which are more equal than others. So we can assume the remaining numbers are less equal and therefore only equal to themselves.

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Other guesses:

Xilpex's rule applies if no digits are zero. If any digit is zero (2 can be written as 02) then there are no equivalents

Because the rules are contrived so I can simply invent whatever I want for the rules that aren't given to me.

Answer 2

20 = no others. same as 0 to 10. because it breaks the following pattern. The green cells are the numbers given and the colored cells is the sum their digits above them. Numbers 0 - 10 break the pattern of the sums as well. Hence the "no others" as they don't follow the pattern of the sums as others

i know there is no 100, 101, i used excel for this, regardless it still doesn't follow the pattern of the sums either way

Answer 3

  0 = no others      ­ 10 = no others      ­ 20 = no others
  1 = no others      ­ 1 1 = 2 9 = 3 1 = 4 9 = 5 1 = 69 = 71 = 89 = 91
  2 = no others      ­ 1 2 = 2 8 = 3 2 = 4 8 = 5 2 = 68 = 72 = 88 = 92
  3 = no others      ­ 1 3 = 2 7 = 3 3 = 4 7 = 5 3 = 67 = 73 = 87 = 93
  4 = no others      ­ 1 4 = 2 6 = 3 4 = 4 6 = 5 4 = 66 = 74 = 86 = 94
  5 = no others      ­ 1 5 = 2 5 = 3 5 = 4 5 = 5 5 = 65 = 75 = 85 = 95
  6 = no others      ­ 1 6 = 2 4 = 3 6 = 4 4 = 5 6 = 64 = 76 = 84 = 96
  7 = no others      ­ 1 7 = 2 3 = 3 7 = 4 3 = 5 7 = 63 = 77 = 83 = 97
  8 = no others      ­ 1 8 = 2 2 = 3 8 = 4 2 = 5 8 = 62 = 78 = 82 = 98
  9 = no others      ­ 1 9 = 2 1 = 3 9 = 4 1 = 5 9 = 61 = 79 = 81 = 99
Delete the tens digit, like follow:

  0 = no others      ­ 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9
  1 = no others      ­ 1 = 9 = 1 = 9 = 1 = 9 = 1 = 9 = 1
  2 = no others      ­ 2 = 8 = 2 = 8 = 2 = 8 = 2 = 8 = 2
  3 = no others      ­ 3 = 7 = 3 = 7 = 3 = 7 = 3 = 7 = 3
  4 = no others      ­ 4 = 6 = 4 = 6 = 4 = 6 = 4 = 6 = 4
  5 = no others      ­ 5 = 5 = 5 = 5 = 5 = 5 = 5 = 5 = 5
  6 = no others      ­ 6 = 4 = 6 = 4 = 6 = 4 = 6 = 4 = 6
  7 = no others      ­ 7 = 3 = 7 = 3 = 7 = 3 = 7 = 3 = 7
  8 = no others      ­ 8 = 2 = 8 = 2 = 8 = 2 = 8 = 2 = 8
  9 = no others      ­ 9 = 1 = 9 = 1 = 9 = 1 = 9 = 1 = 9
So there is no rules to 0,

20 = no others

Answer 4

Some musings below:

The first thing that strikes us as peculiar about rule B is:

That it only applies for numbers from 1 to 3999

Aha! Perhaps the rule is about

Roman numerals?

In particular,

It looks like the number of Is in a particular number's representation is inconsequential.

So, perhaps these sets represent:

The numbers from 0 onwards in order?

To find such a bijection,

Consider I→0, V→1, X→2, L→3, C→4, D→5 and M→6 and add up the letters within a particular number's representation.

This does account for all the equivalences in the rule. However,

Some numbers are missing, such as 54 through 58 which would have a total of (ignoring any Is) LV→3+1=4 so should fit under 19's header (19=XIX→2+0+2=4).

Under this (flawed) approach, the answer to question 2 would then be:

44=XLIV→2+3+0+1=6 so it should fit under 29's equivalence class. Under this interpretation, a full (I think) list of numbers in this class would be 29 through 33 (XXX), 44 through 48 (XLV), 64 through 68 (LXV), 89 through 93 (XC), 109 through 113 (CX), 504 through 508 (DV) and 1000 through 1003 (M).

If a similar approach were to be applied to the original question,

We might be looking for a number system going from 0 to 99 with numbers represented by their distance to the nearest multiple of 20, such that multiples of 20 all have the same non-zero "value", but I can't think of an applicable system off the top of my head.

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Further digging suggests:

The missing values are those with two "multiple of 5 counters", i.e. two of those of VLD. (Perhaps those with more are missing, but we would need to get up to a sum of 9 to confirm.)

Using this reasoning, perhaps:

Those values are in some separate category for some yet-to-be understood reasoning.

In which case the answer to question 2 would be:

44 would be equal to 44 through 48 (XLV), 64 to 68 (LXV) and 504 through 508 (DV).

Answer 5

20 would be:

20 = 20 = 40 = 40 = 60 = 60 = 80 = 80 = 100

Explanation:

The rule (vertically) is: Line 1 + 1, then Line 2 - 1, and so on.