$\textbf{Lucky Seven}$

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 $7$?

$\textbf{Heist}$

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?

$\textbf{Coins on a Table}$

$\textbf{Class Substitute}$

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?

$\textbf{Butcher}$

A clerk at a butcher shop is five feet ten inches tall and wears size $13$ sneakers. What does he weigh?

$\textbf{Mountain}$

Before Mt. Everest was discovered, what was the highest mountain on earth?

$\textbf{Interesting Word}$

Which word in English is always spelled incorrectly?

$\textbf{Picture}$

$\textbf{Race}$

Joe has just passed the person in the second place of a marathon. What position is he in now?

$\textbf{Yolk}$

$\textbf{Haystacks}$

$\textbf{Son}$

$\textbf{The Pet Inc}$

The Pet Inc is owned by three gentlemen pets: a dog, a cat, and a pig. One day, while they are chatting to each other. Mr. Pig says: "Isn't it a bit odd that our surnames match our species, but none of our surnames matches our own species?" The dog replies: "Yes, but does it matter?" Can you relate their surnames and species?

$\textbf{Someone's Name}$

Someone's mother has four sons: North, West and South. What is the name of the fourth son. You are asked to write down the name of the fourth son. What will you write?

$\textbf{Number of Routes}$

The shortest route from point $A$ to $B$ takes $10$ steps. How many such routes are there that do not pass point $C$?

How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$

The $25$ integers from $-10$ to $14,$ inclusive, can be arranged to form a $5$-by-$5$ square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?

In the plane figure shown below, $3$ of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry?

Seven cubes, whose volumes are $1$, $8$, $27$, $64$, $125$, $216$, and $343$ cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units?

What is the median of the following list of $4040$ numbers?

$$1, 2, 3, ..., 2020, 1^2, 2^2, 3^2, ..., 2020^2$$

How many solutions does the equation $\tan{(2x)} = \cos{(\tfrac{x}{2})}$ have on the interval $[0, 2\pi]?$

There is a unique positive integer $n$ such that $$\log_2{(\log_{16}{n})} = \log_4{(\log_4{n})}$$ What is the sum of the digits of $n?$