Is 69 the only nice number?
Mini
9
324
2031
9%
chance

A nice number is a natural number such that its square and cube combined contain all digits exactly once, without needing to use any leading zeros. It is already known that 69 is the only nice number in base ten, and so far, no nice number has been found in any other base either.

This market resolves YES if it is proven that 69 is the only nice number in any base, and NO if it is proven that another nice number exists.

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predicts NO

The quasi-nice stuff (numbers x where x and x^2 are pandigital) makes me pretty confident that there are in fact nice numbers at triple-digit bases, since the current heuristics accurately model their density (first medium, then super low, then super high)

predicts NO

@Conflux I think it's be pretty hard to find one though, so this market will most likely never resolve

@Conflux Yeah, I think they are a pretty strong argument that they exist and we just need to look to higher bases. Only problem is that it will take a very long time to do so if there are no new insights.

predicts NO

@Conflux By the way, what is the highest base that has been fully checked for nice numbers (as far as you know) to date?

predicts NO

@JosephNoonan 48. Highest search I know of is nicenumbers.net (c/o @wasabipesto)

predicts NO

@Conflux I see, so everything up to 44 has been searched for not only nice numbers but also near misses and quasi-nice numbers, while everything up to 48 has been searched for nice numbers, but not necessarily those other types?

predicts NO

@JosephNoonan I think so! Wasabipesto used to have a blogpost on that site describing their process, but I can't find it anymore

@JosephNoonan That's correct! Through base 44 we have information on the niceness distribution and near misses for the entire base range. Through base 48 we checked for nice numbers only. I don't save any information on quasi-nice numbers.

predicts NO

@Conflux @wasabipesto You may also be interested to know that I'm now searching for some new nice number variants: