Question: I have 1000 gold coins and now I'm going to play a game. The game gives me 1000 double coins every day. The rule is like this, I have to bet on the rise and fall of BTC within 15 seconds, and the probability of rise and fall is equal. Every time I invest half of my gold coins. There are 6 multiples to choose from for betting. 1x does not consume double coins, 2x consumes 25 double coins, 3x consumes 50 double coins, 4x consumes 100 double coins, 5x consumes 175 double coins, and 10x consumes 500 double coins. In the case of losing, all invested tokens are lost. In the case of winning, the principal will be returned and an additional reward of multiplying the number of invested tokens by the selected multiple will be obtained. May I ask how to place the best bet? @anoma After detailed expected growth calculations, Grok4 and GPT: For any multiple M, the expected gold coin after each bet is equal to the current gold coin multiplied by [1+(M -1)/4]. Deduction process: Assuming the current gold coin is C, invest B=C/2. Win (probability 0.5): The gold coin becomes C+B × M=C × (1+M/2). Lost (probability 0.5): The gold coin becomes C - B=C/2. Expectation=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 expectation for multiple bets is the product of the initial coins multiplied by the growth factor. Choosing the most efficient M (here 2) and maximizing the number of times will result in the best outcome. They all make me choose twice and keep throwing. I thought I had discovered the mainland, but found out that the daily betting limit was 50000. 😂 I added this again and changed the question to: I have 1000 gold coins, and now I'm going to play a game. The game gives me 1000 double coins every day. The rule is like this, I have to bet on the rise and fall of BTC within 15 seconds, and the probability of rise and fall is equal. Every time I invest half of my gold coins. There are 6 multiples to choose from for betting. 1x does not consume double coins, 2x consumes 25 double coins, 3x consumes 50 double coins, 4x consumes 100 double coins, 5x consumes 175 double coins, and 10x consumes 500 double coins. In the case of losing, all invested tokens are lost. In the case of winning, the principal will be returned and an additional reward of multiplying the number of invested tokens by the selected multiple will be obtained. And the maximum daily betting total cannot exceed 50000 gold coins. May I ask how to place the best bet? What did he do through Monte Carlo simulation (100000 paths) to make me choose 4 times. But this is based on a calculation of 1000 principal, and 500 principal is also the same result, but 50000 principal is the best 10 times. So I think the correct strategy is to choose 4 times on the first day, and the coins will quickly come up later, so you can blindly choose 10 times.