LEMMA 1: All odd positive integers are of type A.

Note that Alice's winning strategy is simply adding one to each number she receives. Note that Bob must begin by writing the number two. Alice can then ensure that Bob will always write an even number by writing Bob's number plus one - this number will be odd, so Bob can only add one (making the number even) or double the number (again making it even). Therefore, Alice can ensure that she will only write odd numbers and that Bob will only write even numbers, meaning that Bob cannot win for odd . Therefore, all odd positive integers are of type A.

LEMMA 2: and are of the same type as .

Note that the player who has a winning strategy for will always have winning strategies for and , as detailed below.

For , this player's winning strategy is as follows: if the other player writes down , they can write down . Since , all moves from here on out must be adding one to the number. Therefore, this player will write down all even numbers larger than , ensuring that they will write . If the other player writes , they can write .

For , this player's winning strategy is as follows: if the other player writes down , they can write down . Since , all moves from here on out must be adding one to the number. Therefore, this player will write down all even numbers larger than , ensuring that they will write . If the other player writes down , they can write down . The other player then must write , ensuring that the original player will be able to write down .

Therefore, the player who has a winning strategy for will always have winning strategies for and , as detailed below.

Using these two lemmas, I claim that a positive integer is of type B if and only if its base-four binary representation consists only of even digits. Clearly, odd numbers will have a base-four binary representation that contains odd digits, making them all type A. Note that any even positive integer can be expressed in the form or for some positive integer . Note that . This means that is the number formed by truncating the last digit of the base-four representation of . If this number ends in an odd digit, then it is odd and the number is of type A. Otherwise, we can continuously repeat the process. The number will only be of type its first digit is and all other digits are even, as otherwise the process can be continued until we reach the first odd digit (reading from the right), making odd and type A. Therefore, a positive integer is of type B if and only if its base-four binary representation consists only of even digits.

By our claim, since the first digit of in base four is , is not of type B, so it is of type A. The least that is type B is , as desired.

Hm if people are interested I could try to post a transcript of my conversation with Champion999 in the comments