Hard puzzles




Hard Discrete Maths

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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Hard Discrete Maths

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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Hard Discrete Maths

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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Hard Discrete Maths

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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Source: Xinfeng Zhou
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Hard Discrete Maths

An optimist and a pessimist are examining a sequence of real numbers. The optimist remarks, ‘Oh jolly! The sum of any eight consecutive terms is positive!’ But the pessimist interjects, ‘Not so fast, the sum of any five consecutive terms is negative.’ Can they both be right? Atmost how large can this sequence be?

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Source: Arthur Engel
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