Suppose Alice is a dishonest bettor, but the casino and all other bettors are honest. Assume the contract$$s balance is currently more than $2000$ ether, we are in Phase $1,$ and no one has won a bet yet.

$($a$)$ Find an attack that can be employed by Alice using which her bet is placed only if it is guaranteed to be a winning bet.

$($b$)$ Find an attack using which Alice can place $10$ winning bets at once, i$.$e$.$ add her address to the winners array $10$ consecutive times.

Problem $2.$ Suppose we are in Phase $1,$ the contract$$s balance is more than $2000$ ether, and the casino has already lost a lot of bets, i$.$e$.$ many bettors have won. Find an attack that can be employed by the casino to ensure no one is paid in Phase $2$ and thus all the money in the contract is eventually funneled back to the casino. Assume that only the casino is dishonest and everyone else is acting honestly. Note that the casino is not willing to lose any money, except for potential gas fees.

Problem $3.$ Suppose we are in Phase $1,$ the contract$$s balance is more than $2000$ ether, Alice has already won $100$ bets and no one else has won anything yet. However, Bob hates Alice and wants to make sure she cannot claim any of her winnings in Phase $2.$ Note that Bob is malicious, so he just aims to hurt Alice$$s earnings as much as possible and does not care if he loses money himself or someone else gets hurt, too. Find an attack that allows Bob to reach his goal. Bob has only $10$ ethers and everyone else is honest.