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.

Currently, the bases up to 48 have been completely searched for nice numbers, with none found, aside from 69 in base 10. You can see https://nicenumbers.net/ for the current status of the serach, as well as @Conflux's last post about it: https://beautifulthorns.wixsite.com/home/post/progress-update-on-the-search-for-nice-numbers

As you can see, it is estimated that there should be about one nice number by the time we get to base 118, but this is only an estimate.

This market resolves once the next nice number is found, provided that all bases smaller than the one it is nice in have been exhaustively checked (or otherwise proven not to have nice numbers). If it is proven that 69 is the only nice number, this will resolve to the last option. If it turns out that somehow the code for the seraches that have been done so far was wrong and there actually is another nice number in a base below 49, then I'll resolve to the "49 - 85" option.