We (meaning the math nerds on Manifold) all know of nice numbers already: Numbers n such that n^2 and n^3 together have all possible digits with no repeats in some base.

But why limit ourselves to hypercubic numbers (i.e., squares, cubes, tesseracts, fifth powers, etc.)? What if we live in Triangland where instead of normal exponents, we use simplex numbers? The simplectic analogue of a nice number would be a number n such that n△2 and n△3, together, have all possible digits with no repeats in a base b (and leading zeros don't count). What do I mean by n△2 and n△3? I'm using n△d as notation for the nth d-simplex number, so n△2 is the nth triangular number, n(n+1)/2, and n△3 is the nth tetrahedral number, n(n+1)(n+2)/6. (If you still don't know what this means, the nth 1-simplex number is just n, and the nth (d+1)-simplex number is the sum of the first n d-simplex numbers). You can think of △ as being the Triangland analogue of the ^ symbol that we use for exponentiation here in Cubeville.

I've checked bases 2 through 20, and assuming I didn't make any mistakes in my math or code, there are no trice numbers in any of those bases. Will I find one by the end of October, or have someone else find one for me? If you find one, tell me the number and base so I can check it.

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