# Problems of the Week

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# 2017-06-12 Basic

You have 3 coins of different weights and a beam balance, as shown above.

What is the minimum number of times you need to use the balance in order to guarantee (in the worst case) that you can sort the coins according to their weights?

Sam suggests that he can make his electric car run a greater distance by installing a windmill on top of it. His theory is that when the car is running, the air blowing against it will help the windmill generate electricity and send it back to the battery.

If it's a calm day and there is no wind around, then will his theory work?

You have a rectangular piece of paper with area 24. Opposite corners are folded behind the paper at 45$^\circ$ angles, as shown below, and the resulting shape is a parallelogram with area 15.

What is the perimeter of the original rectangle?

Robert, the owner of a house called Happy Mansion, was found dead this morning by his butler. The butler called the cops, who then immediately arrested three suspects: Alex, Jenifer, and Joe. Each made a statement:

Alex: "Jenifer is the murderer."
Jenifer: "Joe is not the murderer."
Joe: "Jenifer is not the murderer."

Given that only one of these statements is true and the other two are false, who murdered Robert?

You play a game with a friend with a pile of 20 stones numbered 1~20. You take turns taking 1 stone, 2 consecutive stones, or 3 consecutive stones out of the pile. For example, in your turn, you can take out (19) or (11, 12) or (7, 8, 9) if they are still there. The player who takes the last stone wins.

If you go first, is there a strategy that guarantees you a win?


Clarification: The stones have to be consecutive, but they can be pulled from the middle of the group if you like. For example, on your first move, if you take out three stones, they don't need to be (1, 2, 3). They could, for example, be (11, 12, 13). And likewise for successive turns.

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