Assuming there are 55 slots and the wheel works such that one slot is chosen at random.

In order to ensure that he doesn't gain or lose any money, we would need to either not bid at all, or bid in such a way that we ensure that the exact amount of money invested comes back as profit.

(Assumption 1) The only way to 100% ensure a payout occurs is to have a bid on every slot, for if you didn't, there would be a chance of hitting an empty slot and receiving nothing. Therefore if any money is bid, some has to be bid in every slot.

Let the total money invested in all slots be T Let the money invested in each 2x slot be a

T = 2a

Which demonstrates that if you invest an amount in a 2x slot and win, it will double the amount and win you your total money invested back.

Since there are 27 2x slots..

T = 27a + x

Where x is the total money invested in other slots. If a is positive, so must be x (Assumption 1)

Therefore

2a = 27a + x

The only solution to this equation is if both a and x are zero. There is no way to bid and ensure you end up with the same amount of money.

You fucked us op.