$\textbf{Right to Marry}$

Can a man legally marry his widow's sister in the state of California?

$\textbf{Color the Grid}$

Two geniuses are playing a game of coloring a $2\times n$ grid where $n$ is an odd integer. Each of them in turn picks a uncolored cell and colors it in either green or red until all the cells are filled. At the end of the game, if the number of adjacent pairs with the same color is greater than the number of adjacent pairs with different colors, then the person who picks and colors first wins the game. (An adjacent pair consists of two cells next to each other.) Otherwise, if there are more adjacent pairs with different colors than those with same color, the person who starts later wins. If these two numbers are the same, the result is a tie. Who will win if both players make no mistake?

$\textbf{Split the Coins}$

There are $100$ regular coins lying flat on a table. Among these coins, $10$ are heads up and $90$ are tails up. You are blindfolded and can not feel, see or in any other way to find out which $10$ are heads up. Is it possible to split the coins into two piles so there are equal numbers of heads-up coins in each pile?

$\textbf{Bottle of Bacteria}$

A scientist puts a bacteria in a bottle at exactly noon. Every minute the bacteria divides into two and doubles in size. At exactly $1$ PM the bottle is full. At what time is the bottle half full?

Circle $\omega$ is inscribed in unit square $PLUM$ and poins $I$ and $E$ lie on $\omega$ such that $U$, $I$, and $E$ are collinear. Find, with proof, the greatest possible area for $\triangle{PIE}$.

$\textbf{Label the Boxes}$

There are three boxes, one contains only apples, one contains only oranges, and one contains both apples and oranges. The boxes are incorrectly labeled such that no label identifies the actual contents inside the box. Is it possible to correct all the labels by just randomly retrieving one fruit from one box? You cannot look inside the chosen box.

$\textbf{A Boat Full of People}$

You walk across a bridge and you see a boat full of people, yet there isn’t a single person on board. How is that possible?

$\textbf{Angle on a Clock}$

The time is $3:15$ now. What is the measurement of the angle between the hour and the minute hands?

A group of $100$ friends stands in a circle. Initially, one person has $2019$ mangos, and no one else has mangos. The friends split the mangos according to the following rules:

• sharing: to share, a friend passes two mangos to the left and one mango to the right.
• eating: the mangos must also be eaten and enjoyed. However, no friend wants to be selfish and eat too many mangos. Every time a person eats a mango, they must also pass another mango to the right.

A person may only share if they have at least three mangos, and they may only eat if they have at least two mangos. The friends continue sharing and eating, until so many mangos have been eaten that no one is able to share or eat anymore. Show that there are exactly eight people stuck with mangos, which can no longer be shared or eaten.

$\textbf{Coin Flipping}$

There are $9$ coins on the table, all heads up. In each operation, you can flip any two of them. Is it possible to make all of them heads down after a series of operations? If yes, please list a series of such operations. If no, please explain.

$\textbf{How Far Can You Go}$

There are $50$ motorcycles with a tank that has the capacity to go $100$ km. Using these $50$ motorcycles, what is the maximum distance that you can go?

$\textbf{Shatter the Ball}$

You are in a $100$-story building with two identical bowling balls. You want to find the lowest floor at which the ball will shatter when dropped to the ground. What is the minimum number of drops you need in order to find the answer?

$\textbf{Make Four Liters}$

If you have an infinite supply of water, a $5$-liter bucket, and a $3$-liter bucket, how would you measure exactly $4$ liters of water? The buckets do not have any intermediate scales.

The probability of a specific parking slot gets occupied is $\frac{1}{3}$ on any single day. If you find this slot vacant for $9$ consecutive days, what is the probability that it will be vacant on the $10^{th}$ day?

$\textbf{Coin Toss}$

Joe tosses a coin. If he gets heads, he stops, otherwise he tosses again. If the second toss is heads, he stops. Otherwise, he tosses the coin again. The process continues until either he gets heads or $100$ tosses have been done. What is the ratio of heads to tails in all the possible scenarios?

$\textbf{Three Switches}$

There are three switches in the control room. Two of them are disconnected and the other one is connected to a light in another room. Upon leaving the control room, you will not be permitted to return again. How can you determine which switch is connected to the light?

$\textbf{Bitter Water}$

There are $1000$ bottles of water. All of them are tasteless except one which tastes bitter. How do you find the bottle of bitter water in the smallest number of sips?

$\textbf{Missing Number}$

An $99$-element array contains all but one integer between $1$ and $100$. Find the missing number.

$\textbf{Pirates and Gold}$

Five pirates are trying to split up $1000$ gold pieces according to the following rules

• The $1^{st}$ pirate can propose a plan. If his proposal wins the majority's support (including himself), then it is done. Otherwise, he will be instantly killed.
• If the $1^{st}$ pirate is killed, then the $2^{nd}$ pirate will make a proposal. If his proposal wins the majority's support (a tie does not suffice, i.e. he needs at least $3$ votes), it is done. Otherwise, he will be instantly killed.
• The process continues until a proposal is agreed by a majority.

Assuming all these five pirates are intelligent (i.e. always choose the optimal strategy for himself), greedy (i.e. get as much as gold for himself) and ruthless (i.e. the more pirates dead, the better), what will be the final distribution of the gold?

$\textbf{Child's Name}$

Tracy's mother has four children. The first one is called April, the second is called May, and the third is called June. What is the name of her fourth child?

$\textbf{Defective Machine}$

A company has $10$ machines that produce gold coins. One of the machines is producing coins that are one gram lighter. What is the minimum number of weighs you will need in order to find out the defective machine?

$\textbf{Snail in Well}$

At dawn on Monday, a snail falls into a $12$-inch deep well. During the day, it can climb up $3$ inches. However, during the night, it will fall back 2 inches. On what day can the snail finally manage to get out of the well?

$\textbf{Pirates and Gold (II)}$

What will be the result if all are the same as <myProblem>GetLink/4654</myProblem> except that a proposal only requires $50\%$ of the vote to pass?

$\textbf{Tiger and Sheep}$

One hundred tigers and one sheep are put on a magic island where there is only grass. Tigers on this magic land can eat grass, but they would rather eat the sheep. However, upon having eaten the sheep, the tiger will become a sheep itself. If only one tiger can eat the sheep at any moment, what will happen? The assumption is that all the tigers are intelligent enough to secure their survival first and, if possible, eat the sheep.

$\textbf{Birthday}$

John and Mary know that their boss Joe's birthday is one of the following $10$ dates: Mar $4$, Mar $5$, Mar $8$, Jun $4$, Jun $7$, Sep $1$, Sep $5$, Dec $1$, Dec $2$, and Dec $8$.

Joe tells John only the month of his birthday, and tells Mary only the day.

After that, John first says: “I do not know Joe’s birthday, Mary doesn’t know it either.”

After having heard what John said, Mary says "I did not know Joe's birthday, but I know it now."

After having heard what Mary said, John says "I know Joe's birthday now as well."

What is Joe's birthday?