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$\textbf{Who Finishes the Second}$

Adam, Bob, and Charlie are the only three athletes who are competing in a series of track and field events. The first, second and third places in each event are awarded $X$, $Y$ and $Z$ points respectively, where $X > Y > Z$ and all are integers. It is known that

• Adam finishes first with $22$ points overall
• Bob wins the javelin event and finishes with $9$ points overall.
• Charlie also finishes $9$ points overall.

Who finishes second in the $100$-meter dash and why?

$\textbf{Cookies}$

Steve, Tony, and Bruce have a plate of $1,000$ cookies to share according to the following rules. Beginning with Steve, each of them in turn takes as many cookies as he likes (but must be at least $1$ if there are still cookies on the plate), and then passes the plate to the next person (Steve to Tony to Bruce to Steve and so on). They all want to appear to be modest, but at the same time, want to have as many cookies as possible. This means that they all try to achieve:

1. Have one person get more cookies than himself, and one person get fewer cookies than himself.
2. Have as many cookies as possible.

The first objective takes infinite priority over the second one. If all of them are sufficiently intelligent and can choose the best strategy for themselves, what will be the end result?

$\textbf{Lucky Seven}$

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 $7$?

$\textbf{Heist}$

The chief detective hurries down to the police station after hearing big news: there is a heist at Pi National Bank! The police has brought in seven known gang members seen leaving the crime scene. They belong to the nefarious True/False Gang, so named because each member is required to either always tell the truth or always lie. Although everyone is capable of engaging in wrongdoing, the chief also knows from his past cases that any crime committed by this gang always includes one truth teller. When the chief shows up, he asks the gang members the following questions:

1. Are you guilty?
2. How many of the seven of you are guilty?
3. How many of the seven of you tell the truth?

Here are their responses:

• Person $1$: Yes; $1$; $1$
• Person $2$: Yes; $3$; $3$
• Person $3$: No; $2$; $2$
• Person $4$: No; $4$; $1$
• Person $5$: No; $3$; $3$
• Person $6$: No; $3$; $3$
• Person $7$: Yes; $2$; $2$

After looking these answers over, the chief correctly arrests those responsible gang members. Who out of these seven are arrested?

$\textbf{Guess the Card}$

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?

$\textbf{Class Substitute}$

Kurt, a math professor, needs a substitute for one of his classes today. He sends an email to his three closest co-workers: Julia, Michael, and Mary asking if anyone can help. However, Prof Kurt forgets to give the details of his class. Julia, the department chair, knows which class Kurt teaches, but does not know the time nor the building. Michael plays racquetball with Kurt often, so he knows what time Kurt teaches, but does not know other details. Mary happens to know which building Kurt's class is in, but neither the class itself nor the time.

The possible candidates for Prof Kurt's class are list below.

• Calc $1$ at $9$ in North Hall
• Calc $2$ at noon in West Hall
• Calc $1$ at $3$ in West Hall
• Calc $1$ at $10$ in East Hall
• Calc $2$ at $10$ in North Hall
• Calc $1$ at $10$ in South Hall
• Calc $1$ at $10$ in North Hall
• Calc $2$ at $11$ in East Hall
• Calc $3$ at noon in West Hall
• Calc $2$ at noon in South Hall

After looking over the list, Julia says, "Does anyone know which class it is?" Michael and Mary Ellen immediately respond, "Well, you don't." Julia asks, "Do you?" Michael and Mary Ellen both shake their heads. Julia then smiles and says, "I now know." Which class does Kurt need a substitute for?