$\textbf{Lucky Seven}$

Two non-identical dice both have six faces but do not necessarily have one to six dots on each face. Some numbers are missing and some have more than six dots. These two dice roll every number from $2$ to $12$. What is the largest possible probability of rolling a $7$?

$\textbf{Guess the Card}$

At a work picnic, Todd invites his coworkers, Ava and Bruce, to play a game. Ava and Bruce will each draw a random card from a standard $52$-card deck and place it on their own forehead. So they can see the other's card, but not his or her own. Meanwhile, they cannot communicate in any way. Then they will each write down a guess of his or her own card's color, i.e. red or black. If at least one of them guesses correctly, Todd will pay them $\$50$ each. If both guesses are incorrect, they shall each pay Todd $\$50$. If Ava and Bruce are given a chance to discuss a strategy before the game starts, can they guarantee to win?

After this game, Todd invites two more colleagues, Charlie and Doug, to join a new game. These four players will each draw a card and place it on their own foreheads so only others can see. What is different this time is that instead of color, they should guess the suite, i.e. spade, heart, club, and diamond. If at least one of them makes a correct guess, Todd will pay each of them $\$50$. Otherwise, they should each pay Todd $\$50$. Can these four co-workers guarantee to win if they are given a chance to discuss a strategy before the game starts?

Suppose $a$ and $b$ are both positive real numbers such as $a-b$, $a^2-b^2$, $a^3-b^3$, $\cdots$, are all positive integers. Show that $a$ and $b$ must be positive integers.