I fully expect that it will be the instinct of quite a few to dismiss this challenge as 'too brainy'. But I'm offering the challenge for anyone who wants it.

I'm going to pose some number of riddles or puzzles to you all. You are on your honor not to use anything except your own brain, pencil and paper to come up with solutions. First one to submit a perfect score wins. If no one submits a perfect score within seven days, the person who has submitted the highest score wins. You may only submit the answers once. Warning: Some of these are designed to be really difficult.

Note that more points does not necessarily imply a more difficult task.

Please do not post solutions, but instead use PM to submit solutions.

1. Monty Hall Problem. You are in the final round of a game show, where there a one grand prize hidden behind one of three doors, marked 'Door 1', 'Door 2', and 'Door 3'. The host asks you to select a door, and you select Door 1. The host then opens Door 3, showing you that the prize wasn't behind door 3 [to clarify, he will always show you a non-prize after your first selection]. He now offers to let you switch to Door 2 if you please- or you may keep Door 1.

- (3 points) Is it in your best interest to continue to select Door 1, change to Door 2, or does it not matter? Your solution should provide a reason your answer is correct.
- (7 points) What are the overall odds you win the Grand Prize if you play the strategy mentioned in the answer to the first question?

2. Many Balls, 1 Scale.

- (4 points) You have seven balls, six of which are equal weight, one heavier. You cannot tell by touch which has a different weight. You have a scale (a balance), which you are allowed to use twice. Provide a method for determining which ball is the odd one out.
- (6 points) You have twenty-one balls, twenty of which are equal weight, one heavier. What is the minimum number of uses of the scale/balance that you need?

3. Three Logicians in the Dungeon. Three perfect logicians are thrown into a dark dungeon. They cannot see anything. The executioner walks in, announces that he holds three black hats and two white hats. He places a hat on each of the three logicians, removes the other two hats from the dungeon, and then turns on the lights to the dungeon. Each logician can now see the color hat that the other two are wearing, but not his own- except for the third logician, who is blind. The executioner announces that anyone who can name the color of his own hat will be set free. The first logician announces that he cannot determine the color of his own hat.

- (5 points) The second man announces that he cannot determine the color of his own hat. Can the third man determine the color of his own hat? If so, what color hat is it and why? If not, show why not.
- (5 points) The second man announces that he can determine the color of his own hat. Can the third man determine the color of his own hat? If so, what color hat is it and why? If not, show why not.

4. Two gallons of milk. (4 points) A farmer has two ten-gallon cans of milk, one empty five-gallon can, and one empty four-gallon can. All are unmarked. Wasting no milk, how can he wind up with exactly two gallons of milk in each of the two smaller containers? It is irrelevant how much milk is in each of the larger two containers, so long as the remaining sixteen gallons are all there.

5. Crossing the river. (7 points) Four couples come to a river they must cross over in order to return home. Unfortunately, they only have one boat which holds two-people. In the middle of the stream, there is an island that any number of people may stand on. The men are so extremely jealous that no woman may be left with any other man or men at any time if her own is not present, even in the presence of other women. This applies to the boat, the island, or either shore. Show a method of at most 17 moves of getting them all from shore to shore. (One move is shore to shore, shore to island, or island to shore).

6. A Numbers game and a Card Game. For the following problems, you may assume that each player plays 'perfectly' after your answer- ie, if they can assure themself a win, they will. (Edit for clarity: there are only two players in each game).

- (3 points) Each player, on his turn, adds 1, 2, 3 or 4 to the running count (For example: Player 1 adds 3, P2 adds 4 for 7, P1 adds 4 for 11.. etc). The player that reaches 22 on his turn wins. What must the first player add on his first turn to be assured victory? Why?
- (6 points) A partial pack of sixteen cards - 4 aces, 4 deuces, 4 threes and 4 fours, are placed on the table. On his turn, each player removes a card and adds its value to the running total as in part one. The first player to reach 22 is considered victorious again, but if a player is forced above 22 (think: count is 21, no aces left), that player loses. What rank of card (ace, deuce, three, four) must the first player add on his first turn to assure themself of victory?

A perfect score is fifty. Good luck!

(Again: I'm well aware most of you won't do this because it seems like effort. The inevitable eye-roll aside, get to it, you lazy fucks.)

Edit: NOTE: You may send solutions separately, but you may only send one solution per problem. So you can PM me with the answer #2 and #5, then #3 later, than #4.. etc.