Question: I have 1000 gold coins, and now I want to play a game. The game gives me 1000 doubling coins every day. The rules are as follows: I have to bet on the rise or fall of BTC within 15 seconds, with equal chances for both. Each time, I invest half of my gold coins. There are 6 multipliers to choose from: 1x does not consume doubling coins, 2x consumes 25 doubling coins, 3x consumes 50 doubling coins, 4x consumes 100 doubling coins, 5x consumes 175 doubling coins, and 10x consumes 500 doubling coins. In the case of a loss, the invested tokens are all lost; in the case of a win, the principal is returned, and an additional reward equal to the number of invested tokens multiplied by the chosen multiplier is given. How should I bet for the best outcome? @anoma After detailed expected growth calculations with grok4 and gpt: For any multiplier M, the expected gold coins after each bet = current gold coins × [1 + (M - 1)/4]. Derivation process: Let the current gold coins be C, and the investment B = C/2. Win (probability 0.5): gold coins become C + B × M = C × (1 + M/2). Lose (probability 0.5): gold coins become C - B = C/2. Expected = 0.5 × C × (1 + M/2) + 0.5 × (C/2) = C × [0.5 × (1 + M/2) + 0.25] = C × [0.75 + M/4] = C × [1 + (M - 1)/4]. The expected value of multiple bets is the initial gold coins multiplied by the product of growth factors. Choosing the most efficient M (which is 2 here) and maximizing the number of times will yield the best result. Everyone tells me to keep betting on 2x. I thought I had discovered the mainland, but then I found out the daily betting limit is 50,000. 😂 I added this point and changed the question to: I have 1000 gold coins, and now I want to play a game. The game gives me 1000 doubling coins every day. The rules are as follows: I have to bet on the rise or fall of BTC within 15 seconds, with equal chances for both. Each time, I invest half of my gold coins. There are 6 multipliers to choose from: 1x does not consume doubling coins, 2x consumes 25 doubling coins, 3x consumes 50 doubling coins, 4x consumes 100 doubling coins, 5x consumes 175 doubling coins, and 10x consumes 500 doubling coins. In the case of a loss, the invested tokens are all lost; in the case of a win, the principal is returned, and an additional reward equal to the number of invested tokens multiplied by the chosen multiplier is given. Then the total amount bet daily cannot exceed 50,000 gold coins. How should I bet for the best outcome? He then used Monte Carlo simulation (100,000 paths) and told me to choose 4x. However, this was based on a principal of 1000; the same result applies to a principal of 500, but for a principal of 50,000, 10x is the best. So I think the correct strategy is: choose 4x on the first day, and then the gold coins will quickly increase, allowing for a straightforward 10x thereafter.