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A game you're almost certain to lose...

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If we generalize this problem to other bases, these are what you get.

For Base 2 (binary), there are no solutions.

For Base 4 (quaternary), there are two solutions: 123 and 321.

For Base 6 (senary), there are two solutions: 14325 and 54321.

For Base 8 (octal), there are six solutions: 1274563, 3254167, 5234761, 5614723, 5674321, and 7234561.

I have yet to do the odd bases or the bases higher than 10 (decimal).