This weekend I dropped by the UCFV library and became a community member. The library doesn’t have the biggest selection, but it is cheaper than buying textbooks. Regardless, I picked up a textbook on Abstract Algebra. I didn’t get an opportunity to take an Abstract Algebra course at university, so I thought it would be a fun topic to learn.

Question: Assume that a binary operation on a set has a left unit and satisfies the identity . Prove that is associative and commutative.

Definitions:
Left Unit: where
Commutative:
Associative:

This week’s question has been inspired by the Determinant Game which was posted by ChapterZero. I haven’t determined the heuristics needed to win this game, but it reminded me of similiar AI problems.

Question: The Fort Boyard TV Series had a game where 19 sticks were placed on the ground. Two players would take turns picking up sticks. A player is only allowed to pick up 1, 2, or 3 sticks each round. The person who picks up the last stick loses. What is each player’s “optimal” strategy? If both players play “optimal” strategies, who will win?

When you analysis the game, the solution is trivial. However, you would be surpised how many people got it wrong on national television.

Question: A Koch Snowflake is a fractal which can be built by starting with an equilateral triangle, removing the inner third of each side, building another equilateral triangle at the location where the side was removed, and then repeating the process indefinitely.

Find the area of a Koch Snowflake when the sides of the starting equilateral triangle has the length .

I am finally on Christmas vacation which has gotten me in the Christmas spirit. I thought it would be fun to have a problem related to Christmas. The problem is very easy… which is great because I’m on vacation. In order to solve this problem, you will need to be familiar with the lyrics to the 12 days of Christmas.

Question: If we take the 12 days of Christmas literally, what is the total number of items you will have received at the end of the twelve days? For example, on the first day you have received one item (1 partridge), on the second day you will receive three items (1 partridge, and 2 doves), on the third day you would receive six items (1 partridge, 2 doves, and 3 french hens), etc.

Question: A hemispherical vessel of radius R has a small rounded orifice of area at the bottom. If , how much time does it require to lower the level from to ?