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(while an answer edit brought this to top of active question list) clarify that no other numbers, including 20 and 44, fit into the equivalences shown
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humn
humn
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[Second part and bounty challenge appended in May 2020]


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

         — from [_Animal Farm_](https://en.wikipedia.org/wiki/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.

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

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

  2. 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 nonetheless:

  1. 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.)

[Second part and bounty challenge appended in May 2020]


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

         — from [_Animal Farm_](https://en.wikipedia.org/wiki/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.

**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.

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

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

  2. 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 nonetheless:

  1. 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.)

[Second part and bounty challenge appended in May 2020]


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

         — from [_Animal Farm_](https://en.wikipedia.org/wiki/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.

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

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

  2. 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 nonetheless:

  1. 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.)
(while previous edit's ink is still wet) correct mention of which rule B lines are shown
Source Link
humn
humn
  • 22k
  • 4
  • 60
  • 161

[Second part and bounty challenge appended in May 2020]


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

         — from [_Animal Farm_](https://en.wikipedia.org/wiki/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.

**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 01 through 39 are listed below, accounting for many other eligible numbers as well, with some extra spacing to help distinguish groups of consecutive numbers.

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

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

  2. 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 nonetheless:

  1. 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.)

[Second part and bounty challenge appended in May 2020]


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

         — from [_Animal Farm_](https://en.wikipedia.org/wiki/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.

**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 0 through 39 are listed below, accounting for many other eligible numbers as well, with some extra spacing to help distinguish groups of consecutive numbers.

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

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

  2. 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 nonetheless:

  1. 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.)

[Second part and bounty challenge appended in May 2020]


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

         — from [_Animal Farm_](https://en.wikipedia.org/wiki/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.

**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.

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

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

  2. 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 nonetheless:

  1. 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.)
add second puzzle mainly as hint, use mathjax, add tags
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[Second part and bounty challenge appended in May 2020]

Original puzzle from March 2019:

A contrived simple equivalence rule appliesrule A applies neatly to numbers 0 through 99. but not(Application of this equivalence to any otherfurther numbers is, well, equivocal.) Equivalences ofAll 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.)

­  0 = no others      ­ 10 = no others
  1 = no others      ­ 11 = 29 = 31 = 49 = 51 = 69 = 71 = 89 = 91
  2 = no others      ­ 12 = 28 = 32 = 48 = 52 = 68 = 72 = 88 = 92
  3 = no others      ­ 13 = 27 = 33 = 47 = 53 = 67 = 73 = 87 = 93
  4 = no others      ­ 14 = 26 = 34 = 46 = 54 = 66 = 74 = 86 = 94
  5 = no others      ­ 15 = 25 = 35 = 45 = 55 = 65 = 75 = 85 = 95
  6 = no others      ­ 16 = 24 = 36 = 44 = 56 = 64 = 76 = 84 = 96
  7 = no others      ­ 17 = 23 = 37 = 43 = 57 = 63 = 77 = 83 = 97
  8 = no others      ­ 18 = 22 = 38 = 42 = 58 = 62 = 78 = 82 = 98
  9 = no others      ­ 19 = 21 = 39 = 41 = 59 = 61 = 79 = 81 = 99


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

             ­ 20 = ___ . . . ?

  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.

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

Please use and explain the simplest possible rule,A contrived simple equivalence rule B not purely mathematical,applies neatly to numbers 1 through 3999. that accounts for every(Application of this equivalence from 0 to further numbers is again equivocal.) All equivalences headed by numbers 0 through 9939 are listed below, accounting for many other eligible numbers as well, with some extra spacing to help distinguish groups of consecutive numbers.

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

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

  2. 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 nonetheless:

  1. 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.)

A contrived simple equivalence rule applies neatly to numbers 0 through 99 but not to any other numbers. Equivalences of 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.)

­  0 = no others      ­ 10 = no others
  1 = no others      ­ 11 = 29 = 31 = 49 = 51 = 69 = 71 = 89 = 91
  2 = no others      ­ 12 = 28 = 32 = 48 = 52 = 68 = 72 = 88 = 92
  3 = no others      ­ 13 = 27 = 33 = 47 = 53 = 67 = 73 = 87 = 93
  4 = no others      ­ 14 = 26 = 34 = 46 = 54 = 66 = 74 = 86 = 94
  5 = no others      ­ 15 = 25 = 35 = 45 = 55 = 65 = 75 = 85 = 95
  6 = no others      ­ 16 = 24 = 36 = 44 = 56 = 64 = 76 = 84 = 96
  7 = no others      ­ 17 = 23 = 37 = 43 = 57 = 63 = 77 = 83 = 97
  8 = no others      ­ 18 = 22 = 38 = 42 = 58 = 62 = 78 = 82 = 98
  9 = no others      ­ 19 = 21 = 39 = 41 = 59 = 61 = 79 = 81 = 99


       What would be the entry for 20 in this list?

             ­ 20 = ___ . . . ?

Please use and explain the simplest possible rule, not purely mathematical, that accounts for every equivalence from 0 to 99.

[Second part and bounty challenge appended in May 2020]

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.

**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 0 through 39 are listed below, accounting for many other eligible numbers as well, with some extra spacing to help distinguish groups of consecutive numbers.

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

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

  2. 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 nonetheless:

  1. 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.)
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