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?

@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.

@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?

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

@Conflux NO bettors or zero bettors?

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

@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.

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.

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

@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.

@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)

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

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

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.

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 my financial interest in this is obvious of course.

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

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.

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...

@JoshuaB I suppose if they think they have competition?

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.

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

-69 in base -10 nearly works :P

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

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.

@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.

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)

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]

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]