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


         — from Animal Farm by George Orwell

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.

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    $\begingroup$ Apology for the lack of more specific tags: They would give away the solution. $\endgroup$ – humn Mar 24 at 18:08
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    $\begingroup$ Hurray, a humn puzzle! It's been a while. $\endgroup$ – Rand al'Thor Mar 24 at 19:06
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    $\begingroup$ Is there a way to "watch" a question so that I'm notified of new or accepted answers? I've already starred it. $\endgroup$ – MooseBoys Mar 27 at 1:04
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    $\begingroup$ To be clear, is the relation really only meaningful for numbers 0 thru 99, or are you just saying all numbers outside that range would be "no others"? $\endgroup$ – MooseBoys Mar 27 at 1:07
  • $\begingroup$ Thank you, @MooseBoys. Yes the relation only has relevance to numbers 0 through 99. "No others" would indeed be a great catch-all for other numbers. $\endgroup$ – humn Mar 27 at 12:17
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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.


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.

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  • $\begingroup$ Correct answer, @ferret! But the reasoning is more complicated than necessary. $\endgroup$ – humn Mar 25 at 2:02
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    $\begingroup$ @humn edited with a new "lateral thinking" attempt $\endgroup$ – ferret Mar 25 at 3:04
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    $\begingroup$ You're on the way, @ferret, and gave me an idea for another puzzle. Still missing the essential ingredient. $\endgroup$ – humn Mar 25 at 4:06
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    $\begingroup$ @humn is it because they are rot13 pbagevirq? $\endgroup$ – ferret Mar 25 at 5:31
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    $\begingroup$ Thank you for playing along, @ferret. Pleasure to have met you. $\endgroup$ – humn Mar 25 at 17:06
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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
enter image description here
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

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  • $\begingroup$ Congratulations on the correct answer. And what a wonderfully visual and consistent explanation. Yet the intended explanation is simpler and not nearly as gorgeous. $\endgroup$ – humn Apr 7 at 3:21
  • $\begingroup$ @humn well, the non mathematical explanation would be that numbers with leading or trailing zeroes have no equivalence. 00,01,02,03,04,05,06,07,08,09,10,20 $\endgroup$ – Mel Apr 7 at 9:38
  • $\begingroup$ Your non mathematical 0s approach does fit the pattern, @Mel, but doesn't explain the equivalences as well as your excel solution. $\endgroup$ – humn Apr 7 at 11:33
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My first thought was

The equivalence classes are based on the distance of a number to the closest multiple of 20: $$|11 - 20| = 9, \quad |29 - 20| = 9,\quad |31 - 40| = 9,\quad \ldots$$

However, that did not explain 0 through 10. I could add a 'except for 1 through 10' to my rule, but that wasn't very satisfying.

The second thing I came up with was:

For a number $n$ made of two digits $a$ and $b$, we have $n = a\cdot 10 + b$. If we say those are equivalent to $m=a\cdot 10 - b$, they are recursively equivalent to a lot of numbers. For example, $$97 = 9\cdot 10 + 7$$ $$9\cdot 10 - 7 = 83 = 8\cdot 10 + 3$$ $$8\cdot10 - 3 = 77 = 7\cdot 10 + 7$$ And so on: $97 \rightarrow 83 \rightarrow 77 \rightarrow 63 \rightarrow 57 \rightarrow 43 \rightarrow 37 \rightarrow \ldots$. This shows that numbers with a zero as second digit do not have any other equivalent numbers, as this process would not lead to any new numbers: $$20 = 2\cdot 10 + 0$$ $$2\cdot10 - 0 = 20$$ $20 \rightarrow 20 \rightarrow \ldots$. The only way I managed to explain $1,\ldots,9$ here is to explain you simply can't apply this process to those numbers, as they do not have two digits.

This doesn't work if you accept that 09 is a perfectly fine way of writing 9. So I'm still not very satisfied with this solution.

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  • $\begingroup$ M-ou-se, the indended explanation is hidden inside your second approach (about multiples of 10). That is the closest any has yet gotten! Still, the intended solution is simpler and not quite as mathematical. $\endgroup$ – humn Apr 7 at 11:42
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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.

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    $\begingroup$ Thank you for taking the bait, Xilpex. Not quite the solution, though. For instance, it doesn't explain the entry for 10. $\endgroup$ – humn Mar 24 at 18:23
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    $\begingroup$ @humn Ok. I'll see if there is any other answer... :D $\endgroup$ – Xilpex Mar 24 at 18:25
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    $\begingroup$ Plus there is no $100$. $\endgroup$ – Arnaud Mortier Mar 24 at 18:36
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  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

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    $\begingroup$ Keep going, @user58107! It's simpler than that. $\endgroup$ – humn Mar 25 at 6:42
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    $\begingroup$ look the column, my English very poor, can't explain clarification. $\endgroup$ – user58107 Mar 25 at 6:46
  • $\begingroup$ Oh, oh oh oh, @user58107, this puzzle relies on English. (Big give-away.) Thank you for hitching the ride. $\endgroup$ – humn Mar 25 at 6:52
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    $\begingroup$ @humn Maybe a language tag then? $\endgroup$ – Rubio Apr 6 at 4:07
  • $\begingroup$ Right, @Rubio, language tag added. I was trying to not give away that aspect but did in the comment above and the time is ripe anyway. $\endgroup$ – humn Apr 7 at 3:11

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