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?

Dyson's Double

Is there an integer such that, if you take its last digit and move it to the front, its value doubles?

Reportedly solved by Mr. Dyson in two seconds (spoiler).

Mathematical Constant Day

Celebrating "Pi Day" on March 14 seems pretty arbitrary, to say nothing of lame. When are you supposed to observe "e Day" or "Phi Day"? If you're going to pick a point during the year to recognize pi, why not the moment between January 1 and December 31 proportional to a circle's diameter's ratio to its circumference? This will fall on April 26 most years, and I suppose should be called "One Over Pi Day." (Alas, I see this has all been covered.)

On Pi Day, someone must observe that "The first 144 digits of pi add up to 666." What are the chances that a randomly generated series of consecutive digits will at some point add up to 666 (or any other large, evil number)?

The Hardest Logic Puzzle Ever

Three gods A, B, and C are called, in no particular order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for yes and no are 'da' and 'ja', in some order. You do not know which word means which.

Prime directive

Which bases, if any, is the number 10101 prime in? Prove your answer.

Integer bombing

A ship is sailing at a constant, integral speed along the number line. At time = 0 it is located at position 100, but you do not know its speed or direction. Starting at time = 1 second you can begin dropping one bomb every second at locations you specify in advance. Can you be guaranteed of hitting the ship?