Discrete Maths puzzles

A group of 5 people want to keep their secret document in a safe. They want to make sure that in future, only a majority (>=3) can open the safe. So they want to put some locks on the safe, each of the locks have to be opened to access the safe. Each lock can have multiple keys; but each key only opens one lock. How many locks are required at the minimum? How many keys will each member carry?

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We have a beam balance (with two pans to compare weights) and a positive integer N. How do we select fewest number of pebbles to weigh all possible integers from 1 to N

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Several gas stations on a circular trek have between them just enough gas for one car to make a complete round trip. Prove that if you start at the right station with an empty tank you shall be able to make it all the way around.

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Source: P. Winkler
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Can you pack 53 bricks of dimensions 1x1x4 into a 6x6x6 box? The faces of the bricks are parallel to the faces of the box

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Solution
Source: Xinfeng Zhou
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There is a crazy clock in Alice's Dream, it has two hands initially pointing at 12. The minute hand moves clockwise, making 5 rounds (with varying speeds) and comes back to 12. In the same time, the hour hand goes anti-clock wise, finishing 4 rounds and returns to 12. How many times did the two cross each other ? (Cross means meet & pass through, hence ignore start & end)

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