Will I learn of a "nice number" besides 69 by the end of March 2023?
Resolved
N/A

69^2 = 4761

69^3 = 328509

As you can see, the square and cube of 69, in base 10, contain all the digits with no repeats. In general, a nice number, in some base, is one where the square and cube, written in that base, contain all the digits in that base with no repeats.

[EDIT 12/27/22: -69 in base 10 has been brought to my attention. It technically meets my definition, but is not in the spirit of the market. I'm still thinking about how to handle this and am temporarily halting trading.]

[EDIT 12/28/22: I've decided to resolve N/A and create a new market with an improved definition, linked below. See the comments for more discussion of this decision.]

So far, this is the only nice number I know of. I suspect that there are an infinite number, but they only start appearing around base 120 to 130; I checked up to around base 30.

I discuss this more in my blog post http://tinyurl.com/confluxblog/post/is-69-unique.

Anyway, hopefully this market inspires you to do some research into finding nice numbers, or to find someone else online who has!

I may also provide mana rewards for partial results.

Close date updated to 2022-12-27 12:23 pm

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Conflux avatar
Confluxpredicted NO at 26%

In light of the -69 situation: My current plan is to resolve N/A, since there are compelling arguments for both counting -69 in base 10 as a nice number (which causes a YES resolution) and not counting it (which causes the market to remain open). However, I want to wait a bit longer (maybe a day?) if anyone has any better suggestions.

The argument for counting it (resolving YES): -69 in base 10 (which @Cadence asked me about) simply meets the resolution criteria. Its square and cube, 4761 and -328509, contain all digits with no repeats. (69i and -69i, amusingly, also work.)

The argument for not counting it (remaining open): I created this market to encourage research into nice positive, real integers in higher positive integer bases, and essentially all bettors treated the market as if it was about this. -69 is a degenerate example, and if I had thought of it, I would have excluded it. Some statements in my blog post, like that 69 is the only nice number in base 10, clearly suggest I wasn’t looking at negative numbers. But I didn’t rule them out.

I think this is a classic dispute between the exact wording of the resolution criteria and the clear spirit of the market, and I want to act as honorably as possible. I think N/A (cancelling the market and reverting all trades), followed by making a new market with better resolution criteria, is probably the best. What do you all think?

AM avatar
AMpredicted YES at 26%

@Conflux Has anyone so far placed bets assuming degenerate examples like -69 would count? If not then I see no reason to not just continue with this market and reword the description to explicitly exclude negative or non-real solutions.

MichaelWheatley avatar
Michael Wheatley

@Conflux FWIW, as a disinterested bystander, I want you to ignore -69, on the grounds that it's obviously not what the market is asking, and cancelling/recreating markets because of small technicalities like this is a lot of added overhead and annoyance. Are there any actual good faith -69 bettors out there to be disappointed by this?

Conflux avatar
Confluxpredicted NO at 26%

@MichaelWheatley @AM I believe that no bettors were considering these degenerate examples.

MichaelWheatley avatar
Michael Wheatley

@Conflux NO bettors or zero bettors?

Conflux avatar
Confluxpredicted NO at 26%

@MichaelWheatley Zero bettors, haha. Great example of ambiguity in the wild!

JoshuaB avatar
Joshuapredicted YES at 26%

@Conflux I personally think that this should resolve either YES or N/A (and am leaning towards N/A)

Why should it resolve to YES? Because the market description outlays what a nice number is, and -69 falls under that definition of what a nice number is, so @Conflux has indeed learned of a nice number besides 69 before March 2023.

Why should this resolve to N/A? Well, first off it seems kinda unfair to all the NO bettors to have it resolve YES based on this little technicality. Not only that, but if @Cadence had actually bet a bunch of money on YES because of the technicality, then it seems to me that leaving the market open is a really terrible option. And even though they didn’t, it seems like a poor precedent for market resolution to depend on other people’s bets.

IsaacKing avatar
Isaac Kingpredicted YES at 26%

If anyone had actually placed bets on YES due to knowing about -69, or due to a general realization that there could be an answer outside the bounds of what other people were expecting, then this should definitely resolve YES.

Given that it doesn't seem that happened, and all YES traders are ok with it, I think it would be fine to edit the description to have the correct resolution criteria and reopen the market.

levifinkelstein avatar
levi finkelsteinpredicted YES at 26%

@IsaacKing Are we going to ask all the YES traders if they are ok with it?

IsaacKing avatar
Isaac Kingpredicted YES at 26%

@levifinkelstein I think Conflux already asked the big ones, and all the others have the option to speak up now.

Conflux could also just directly repay any who aren't happy with it.

JoshuaB avatar
Joshuapredicted YES at 26%

@IsaacKing I am a YES bettor that thinks it should resolve YES or N/A (although I did not bet with degenerate cases like -69 in mind)

BoltonBailey avatar
Bolton Bailey

I'll add my voice to @MichaelWheatley 's as another disinterested bystander who feels the question was clearly intended to be about natural numbers.

WilliamHoward avatar
Will Howard

@Conflux I'm also disinterested and also agree that -69 definitely shouldn't count

Conflux avatar
Confluxpredicted NO at 26%

Thanks everyone for your opinions! I intend to move forward with N/A.

I am definitely influenced by the fact that reopening seems to be the consensus answer. In the future, I’m leaning towards establishing an official policy where I override my resolution criteria in rare situations like this, when circumstances emerge that I didn’t write them to handle. If I had such a policy, I would reopen. But I don’t, so I feel committed to not totally ignoring my resolution criteria.

AMS avatar
Adampredicted NO at 26%

I would ignore -69, on the basis that there's a fairly narrow interpretation ("a natural number in a standard integer base ≥2") that captures the interesting question while excluding weird edge cases.

Sometimes "you know what I mean" doesn't work well in math, because the edge cases are where the action is, but I think the action here is in the originally intended (narrow) interpretation.

BTW, if you agree, it's probably best to preemptively rule out non-integer bases, bases that don't follow the standard rules, and non-integer nice numbers. I don't know there to be additional edge cases there but there could be.

AMS avatar
Adampredicted NO at 26%

@AMS my financial interest in this is obvious of course.

JoshuaB avatar
Joshuabought Ṁ4 of YES

There are no nice numbers in base 34 or below! (save 69 in base 10 of course)

AMS avatar
Adambought Ṁ50 of NO

Numbers equivalent to 4 mod 5 would be the place to look, since they have a much wider range. If my math is right, there's (heuristically) a ~50% chance that the first base with a nice number above 69 is one of 99 or 104.

99^99/99! is about 4x10^41 so this is gonna be hard. Also, the middle digits of products of large numbers are very unstructured, to the point that "square and take the middle digits" is a famous pen-and-paper PRNG algorithm. I'm not a number theorist, but I'd be surprised if it was tractable to find an explicit number of this form.

JoshuaB avatar
Joshuabought Ṁ100 of YES

I wonder if there’s any reason (save honor) for if someone were to hypothetically find a nice number to immediately tell @Conflux... seems to me, in that hypothetical, that they should wait till late March when the market is way lower...

Conflux avatar
Confluxpredicted NO at 54%

@JoshuaB I suppose if they think they have competition?

JoshuaB avatar
Joshuapredicted YES at 55%

Ooh here’s a cool find:

18^3 = 5832

18^4 = 104976

Base 10 is the only base between base 2 and base 32 where there exists a number with that property.

Conflux avatar
Confluxpredicted NO at 55%

@JoshuaB really a special base we've got here! we should use it for stuff

Adam avatar
Adampredicted YES at 69%

-69 in base -10 nearly works :P

Adam avatar
Adampredicted YES at 69%

@Adam wait no that's not how negative bases work anyways

Yev avatar
Yev

I tried looking for numbers where x and x^2 contain all the digits (instead of x^2 and x^3).

The number must have roughly base/3 digits instead of base/5, and the break-even point is at 11.5 instead of 130.5. I ran it up to base 19 inclusive:

Base 6: 32^2 = 1504

Base 8: 256^2 = 73104

Base 9: 615^2 = 420837

Base 17: [ 2 10 11 3 14 16]^2 = [ 6 15 5 9 8 7 0 12 13 4 1]

Base 17: [ 3 11 7 15 14 6]^2 = [13 8 9 1 4 10 5 16 12 0 2]

Base 18: [ 5 4 10 12 13 6]^2 = [ 1 9 11 2 3 7 16 8 17 15 14 0]

Base 18: [ 5 8 3 9 7 2]^2 = [ 1 11 13 12 6 17 15 0 16 14 10 4]

Base 18: [ 5 8 16 15 10 13]^2 = [ 1 12 3 14 9 11 4 2 6 0 17 7]

Base 18: [ 6 10 13 15 8 16]^2 = [ 2 7 9 12 5 14 11 17 3 1 0 4]

Base 18: [10 1 3 16 14 7]^2 = [ 5 11 6 8 2 12 15 4 9 17 0 13]

Base 18: [10 9 15 5 16 12]^2 = [ 6 3 4 7 11 8 17 13 2 1 14 0]

Base 18: [10 14 15 1 17 11]^2 = [ 6 9 3 0 7 5 16 12 4 8 2 13]

Base 18: [11 5 8 2 9 16]^2 = [ 7 1 13 10 15 3 6 0 12 17 14 4]

Base 18: [13 6 1 8 10 12]^2 = [ 9 15 16 3 7 4 11 5 2 17 14 0]

Just as the probabilistic argument predicts, we have a few solutions for small bases, then a big gap, and then lots of solutions for large bases. But for some reason the gap ends later than predicted; I don't know if that's significant.

Conflux avatar
Confluxpredicted NO at 75%

@Yev Fascinating! I wonder if the implication that it’s less likely than pseudorandom (maybe because of last digit stuff?) carries over to x^2 and x^3.

Yev avatar
Yev

And with leading zeros, up to base 19:

Base 4: 03^2 = 21

Base 10: 0567^2 = 321489

Base 10: 0854^2 = 729316

Base 13: [ 0 7 1 8 2]^2 = [ 3 11 9 12 10 5 6 4]

Base 13: [ 0 10 12 6 2]^2 = [ 9 3 1 5 8 7 11 4]

Base 16: [ 0 4 14 10 11 5]^2 = [ 1 8 2 12 13 7 6 3 15 9]

Base 16: [ 0 11 1 6 4 13]^2 = [ 7 10 14 12 8 15 5 3 2 9]

Base 16: [ 0 12 5 6 11 10]^2 = [ 9 8 3 14 13 1 7 15 2 4]

Base 18: [ 0 2 14 11 15 5 3]^2 = [ 7 16 10 4 13 1 17 6 8 12 9]

(no base 19, even though it has a good remainder mod 3)

Yev avatar
Yev

All base 20s, without leading 0s:

Base 20: [ 1 13 18 6 15 19 3]^2 = [ 2 17 10 7 5 0 4 11 12 8 16 14 9]

Base 20: [ 1 14 6 8 11 5 12]^2 = [ 2 18 17 19 3 13 9 10 0 16 15 7 4]

Base 20: [ 1 17 8 4 10 11 6]^2 = [ 3 9 19 12 2 15 5 14 18 0 7 13 16]

Base 20: [ 1 18 10 12 2 7 6]^2 = [ 3 14 4 11 13 9 8 19 0 15 17 5 16]

Base 20: [ 2 0 8 17 10 5 14]^2 = [ 4 1 15 13 19 18 7 6 3 11 12 9 16]

Base 20: [ 2 3 0 10 16 19 7]^2 = [ 4 12 11 6 13 5 1 15 14 17 18 8 9]

Base 20: [ 2 19 13 17 12 16 15]^2 = [ 8 18 3 7 14 9 1 10 4 6 0 11 5]

Base 20: [ 3 7 5 0 13 1 8]^2 = [11 6 2 15 12 17 18 14 10 16 9 19 4]

Base 20: [ 3 12 18 9 7 8 19]^2 = [13 5 17 16 10 14 4 15 0 11 6 2 1]

Base 20: [ 4 5 13 15 6 14 19]^2 = [18 7 2 9 16 17 12 3 8 0 11 10 1]

Yev avatar
Yevpredicted NO at 62%

Base 21:

  1. [ 4 20 6 2 14 15 9]^2 = [ 1 3 13 19 16 11 10 12 5 8 17 7 0 18]

  2. [ 5 15 13 19 7 10 16]^2 = [ 1 12 0 6 11 9 14 2 18 20 8 3 17 4]

  3. [ 7 11 19 0 12 17 4]^2 = [ 2 15 5 9 3 8 20 18 14 1 6 13 10 16]

  4. [ 8 0 19 6 11 20 10]^2 = [ 3 1 14 15 17 18 12 7 4 13 2 9 5 16]

  5. [ 8 15 7 1 2 6 10]^2 = [ 3 13 4 12 0 14 20 9 17 11 5 18 19 16]

  6. [ 8 16 20 5 1 10 9]^2 = [ 3 14 12 2 13 7 19 17 6 4 11 0 15 18]

  7. [ 8 18 6 10 9 7 11]^2 = [ 3 15 14 19 4 0 20 17 5 13 1 2 12 16]

  8. [10 1 12 15 9 13 14]^2 = [ 4 17 11 5 6 8 3 0 19 20 2 18 16 7]

  9. [11 20 7 14 1 15 2]^2 = [ 6 17 5 16 10 8 19 9 3 12 13 0 18 4]

  10. [12 1 8 17 20 3 9]^2 = [ 6 19 13 4 11 10 5 7 0 16 2 14 15 18]

  11. [12 1 18 9 13 17 4]^2 = [ 6 20 3 5 11 19 15 7 2 14 8 0 10 16]

  12. [13 0 19 14 17 16 6]^2 = [ 8 2 3 9 5 10 11 1 18 12 20 7 4 15]

  13. [13 20 3 10 17 8 19]^2 = [ 9 5 18 15 2 16 12 1 0 7 14 11 6 4]

  14. [14 1 3 17 19 5 11]^2 = [ 9 8 12 4 6 2 0 20 13 15 18 7 10 16]

  15. [14 4 17 18 19 7 6]^2 = [ 9 13 10 20 16 5 11 3 2 0 12 8 1 15]

  16. [14 7 11 6 2 4 10]^2 = [ 9 17 3 15 20 1 19 8 13 5 12 18 0 16]

  17. [14 13 19 2 18 16 3]^2 = [10 4 20 15 8 1 0 7 6 5 17 11 12 9]

  18. [15 13 20 1 3 17 16]^2 = [11 14 7 19 9 2 8 5 0 18 6 12 10 4]

  19. [16 20 11 8 12 14 9]^2 = [13 15 5 10 2 19 7 0 6 1 17 4 3 18]

  20. [17 7 20 10 15 13 12]^2 = [14 8 1 5 0 11 2 16 6 9 4 19 3 18]

  21. [17 16 6 13 12 11 14]^2 = [15 1 0 9 5 3 4 8 18 19 20 10 2 7]

  22. [18 0 17 16 14 12 13]^2 = [15 10 9 11 7 2 6 4 3 8 20 19 5 1]

  23. [18 0 20 6 4 8 14]^2 = [15 10 13 17 12 3 16 5 1 11 9 19 2 7]

  24. [18 4 11 13 5 14 20]^2 = [15 16 17 19 10 6 2 7 9 0 8 3 12 1]

  25. [18 20 14 1 12 19 5]^2 = [17 3 8 10 0 13 15 11 16 9 6 7 2 4]

  26. [19 1 13 12 10 16 18]^2 = [17 6 20 15 7 11 8 0 4 2 5 14 3 9]

  27. [19 3 6 8 13 20 16]^2 = [17 10 0 2 12 15 14 9 11 5 18 7 1 4]

  28. [19 6 7 18 20 16 3]^2 = [17 15 13 5 0 8 11 14 1 4 10 2 12 9]

  29. [19 10 8 11 12 9 5]^2 = [18 2 1 13 6 0 14 15 3 17 20 16 7 4]

  30. [19 13 6 20 12 10 5]^2 = [18 7 11 2 0 8 15 1 3 16 9 14 17 4]

  31. [19 17 13 4 1 5 10]^2 = [18 15 12 14 2 3 0 6 9 11 8 7 20 16]

Yev avatar
Yev

In base 6:

5^2 = 041

5^3 = 325

Conflux avatar
Confluxpredicted NO at 77%

@Yev Ooh, I like this! I declare it semi-nice.

Conflux avatar
Confluxpredicted NO at 77%

Seems like this is the only small example that's only missing one digit, very convenient that that digit is 0 so you can write it leading like that!

8 avatar
Trongbought Ṁ22 of YES

base 10,

19^2 = 361

19^3 = 6859

hope im not misunderstanding something

Conflux avatar
Confluxpredicted NO at 72%

@Trong 6 is repeated, and you're missing 0, 2, 4, and 7

8 avatar
Trongpredicted YES at 72%

@Conflux oh i see