Navy Rat Snail lottery

A peculiar lottery is played in an eastern country. Three tokens are randomly drawn (without replacement) from a bag of fourteen tokens. Each token has a unique mark, one of the national symbols of the region:

Navy
Milk
Ogee
Pot
Quill
Rat
Snail
Tuber
Uvala
Vole
Whelp
Xylem
Yarn
Zadar

Players enter the lottery by buying tickets and choosing three symbols for each ticket. A ticket wins if at least two of the three symbols guessed match the symbols on the tokens drawn from the bag.

What is the minimum number of tickets necessary to guarantee a prize?

Rencontre

In October of 1779, Euler prepared for the St. Petersburg Academy a paper on “a curious question from the doctrine of combinations.” So great was the backlog of Euler's writings that this was not published in the Academy's Memoirs until 1811, more than a qurter-century after his death. This delay notwithstanding, he provided an example not only of a “curious” problem but also a method of attack—[redacted]—that has become one of the primary weapons in the combinatorial arsenal.

The problem had been posed, and solved, decades earlier. In 1708, Pierre Remond de Montmort (1678-1719) discussed a card game called Treizeor, more descriptively, Rencontre (“coincidence”). The rules are these: a player shuffles a deck of 13 cards from ace to king, then deals them one at a time saying “one” with the first card dealt, “two” with the second, and so on. Victory is achieved if at some point the card overturned exactly matches the word spoken—for instance, if the fourth card turned over is a “four” or the eleventh card laid down is a “jack.” should such a coincidence occur, the dealer wins.

Montmort determined the probability of winning thereby earning for himself a place of honor in the history of combinatorics....

—William Dunham, Euler: The Master of Us All

Born on a Tuesday

Suppose I tell you that I have two children and one of them is a boy, what is the probability that I have two boys?

The correct answer is not 1/2 but 1/3. How can that be? Well there are four possible combinations, BB, GG, BG and GB but but at least one is a boy so you can get rid of GG. So all that's left is BB, BG and GB; and in only one of those three possibilities do I have two boys.

Now I tell you that I have two children and one of them is a boy born on a Tuesday. What is the probability that I have two boys?

Wise Men Don't Wear Hats

An emperor decides to give 100 of his philosophers a test. The philosophers will stand in line, one behind the other, so that the last person in the line sees everyone else. The emperor has 101 hats, each of a different color, and the philosophers know all the colors. The emperor puts all but one of the hats on the heads of the philosophers, one per philosopher. The philosophers can only see the colors of the hats on people in front of them. Then, in any order they want, each philosopher guesses the color of the hat on his own head. Each hears all previously made guesses, but other than that, the philosophers cannot speak. They are not allowed to repeat a color that was already announced. Each person who guesses his color wrong will get his head chopped off. If a color is guessed a second time, all the philosophers will have their heads chopped off. The ones who guess correctly go free. The rules of the test are given to them one day before the test, at which point they have a chance to agree on a strategy that will minimize the number of people who die during this test. What should that strategy be?

Bug march

A bug starts at the top-right corner of a square and marches counterclockwise around the perimeter of the square at a uniform speed. A spider starts at the bottom-right corner of the square and moves at the same speed as the bug, always in the direction of the bug. Where is the spider when the bug reaches the top-left corner? Does the spider catch the bug?

The penny drops

Stolen from a place. What is the shape of a coin that has probability 1/3 of landing (and standing) on its edge?

Naively, you accept

While walking through the park one day you encounter a man you
have never met before, who comes up to you and shows you a
folded up piece of paper.

"On this piece of paper I have written an integer from
0 to 119," he says. "If you give me $10, I will allow you to choose one of the
groups I am about to list; if the number I wrote is in the group you
pick, I will give you $30. If not, I will give you nothing.
Each group contains 60 of the 120 numbers from 0 to 119."

Naively, you accept. He lists the groups as follows:

1. The numbers from 0-59
2. The numbers from 60-119
3. Even numbers
4. Odd numbers
5. Numbers which have a remainder of 0 or 1 when divided by 4
6. Numbers which have a remainder of 2 or 3 when divided by 4
7. Numbers which have a remainder of 0, 1, or 2 when divided by 6
8. Numbers which have a remainder of 3, 4, or 5 when divided by 6
9. Numbers which have a remainder of 0, 1, 2, or 3 when divided by 8
10. Numbers which have a remainder of 4, 5, 6, or 7 when divided by 8
11. Numbers which have a remainder of 0, 1, 2, 3, or 4 when divided by 10
12. Numbers which have a remainder of 5, 6, 7, 8, or 9 when divided by 10

Which group do you choose?

More interestingly, what is your strategy for playing multiple iterations of this game?