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

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 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.

@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.

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.

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.

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.

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.

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:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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!