One day, I wanted to be able to represent all numbers less than 20 without erasers or computers (Don't ask, OK?) I immediately thought of making cards1. For example, in base 10, you need 2 for the tens digit (saying 1,2), when there is no tens digit, don't put a card) and 10 for the ones digits, saying 1-9. However, that's a lot of cards! The smallest number of cards needed is 8, for base 3.
1 The cards are one sided, so no flipping cards around
In fact, most numbers have base 3 as the least amount of cards needed. Can you find the smallest 5 numbers, excluding the trivial examples 1 and 3 where "number" is like the 20 in the example (note 20 is not an answer)that has a base other than 3 that requires less cards?
A formula for cards needed in base b and numbers up to n is:
(First digit of n in base b)+(b*length of n in base b)-b.
To clarify, no rearranging cards. Tens digits cards can't go to the ones digits, for example.