Hello, foreverbrokenpromises!

I found the answer by experimenting . . .

The number of possible bases for which is a perfect square is:

. .

The number is equal to: .

. .which is already a perfect square!.

Since the number uses {1, 2, 3}, the base must be at least 4.

Therefore, is a perfect square forbase . . . answer (e)any

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Examples:

. .