$\textbf{Interesting Locations}$

$\textbf{Lighting Bulb}$

There are $100$ bulbs, all are off, each of which is controlled by a switch. Joe was playing with them in the following way:

• In the first round, he toggled every switch. So, all the lights are on now.
• In the second round, he toggled switches $2$, $4$, $6$, $\cdots$, $100$. Now half are on and half are off.
• In the third round, he toggled switches $3$, $6$, $9$, $\cdots$, $99$,
• $\cdots$
• In the $10^{th}$ round, he toggled switch $10$, $20$, $\cdots$, $100$
• $\cdots$
• In the $100^{th}$ round, he toggled the switch $100$

Now, the question is, how many bulbs are on at the end?

$\textbf{Boys v.s. Girls}$

In a remote town, people generally prefer boys over girls. Therefore, every married couple will continue giving birth to a baby until they have a son. Assuming there is fifty-fifty chance for a couple to give birth to a boy or a girl, what is the ratio of boys to girls in this town over many years?

$\textbf{Bus Direction}$

Which way is this moving bus going?

$\textbf{Offer Letter}$

After a whole day of interviews, a HR manager comes with three sealed envelopes. One of them contains an offer letter, and the other two contain rejection letters. You can select one of them and will be hired if you get the offer letter. After you pick one envelope, the HR manager opens one of the other two which contains a rejection letter and offers you a chance to change your mind. Should you change your selection? Explain.

$\textbf{Average Speed}$

Joe travels at an average of $30$ miles per hour from home to visit a friend who lives $60$ miles away. How fast should he drive on his way straight back to home so that his average speed is $60$ miles per hour for this entire trip?

$\textbf{Mafia}$

You are captured by a mafia. He puts two bullets in adjacent chambers of a standard $6$-chamber revolver. Then he points the gun at your head, and pulls the trigger. You survives. He thinks you may be a lucky man and thus promises to free you if you can survive the second shot. Meanwhile, he also gives you the option to re-spin the revolver before he pulls the trigger again. Should you accept his offer?

$\textbf{Red Cards}$

There are $7$ cards. Two of them have both sides red, two of them have both sides black, the rest three have one side red and one side black. Joe draws one card randomly and finds one side is red, what is the probability that the other side is red too?

$\textbf{Number of People}$

There are $10$ people dancing in a room with only one door. A terrorist busts in and opens fire. Eight people are killed immediately. How many people are there inside the room now? 

$\textbf{Stack of Coins}$

Can you fit a stack of quarter coins as high as the Empire State Building into a normal room?

$\textbf{Children's Age}$

Joe and John meet at a bar and become acquainted. John tells Joe that he has three children. So Joe asks John "How old are they?". "Well, the product of their ages is $72$.", John replies, "and the sum of their ages is the street number of this bar." Joe walks out the door, checks the number, and comes back "The information is not sufficient." John says "I agree. Here is another piece of information: my youngest child likes ice cream." Joe thinks for a minute and says "Now I know your children's ages."

What are the ages of John's children?

$\textbf{Tricky Sequence}$

Find the next term in the sequence: $F21$, $S23$, $T25$, $T27$, $S29$, $M31$.

$\textbf{Medalists}$

Five runners, $A$, $B$, $C$, $D$, and $E$, enter the final. The fastest three win a gold, silver, and bronze medal, respectively. The other two get nothing. Who are the three medalists if all of the following statements are false?

• $A$ does not win the gold and $B$ does not get the silver.
• $B$ does not get the bronze and $D$ does not win silver.
• $C$ wins a medal, but $D$ does not.
• $A$ wins a medal, but $C$ does not.
• Both $D$ and $E$ win a medal.

$\textbf{Orange}$

I am an honest man. I can tell you that I love and hate orange at the same time. Do you know why?

$\textbf{My Name}$

$\textbf{Three Coins}$

With an infinite supply of coins, what is the minimum number of coins required for each and every coin to touch exactly three other coins.

$\textbf{Circular Killing}$

One hundred prisoners on the death row are ordered to stand in a circle and are numbered from $1$ to $100$ in sequence. The king then gives a sword to No $1$. No 1 kills the No $2$ and passes the sword to No $3$. The No $3$ then kills No $4$ and passes the sword to the next person alive, i.e. No $5$. All people continuously does the same until only one person survives. Who is the last survivor?

$\textbf{Who Finishes the Second}$

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?

$\textbf{Birthday Problem}$

Statistically what is the minimum number of people among which the probability of two people having the same birthday exceeds $50\%$? How about if this probability needs to exceed $99.9\%$?

$\textbf{Cookies}$

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 $7$?

$\textbf{Heist}$

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?