A very interesting riddle for those of you with some basic background in Set Theory.

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A very easy riddle. Four frogs are sitting on the corners of the unit square (i.e. they have coordinates (0,0), (0,1), (1,1) and (1,0) ). Each turn, a frog can jump over any other frog, thereby transferring itself to the symmetrical point on the other side of the static frog. For example, if the frog at (0,0) jumps over the frog at (1,1) it will land on (2,2).

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 This is a cute puzzle. Consider an infinite checkerboard divided in two with an infinite line lying along the x-axis, as depicted below:

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Dany Valevsky gave me this very cool riddle.

You are given a vector of size N, the elements of which are numbers in the range 1,…,N-1. I.e. there is at least one repeating element. Give an algorithm that finds a repeating element (it does not matter which one, in case there are several) with O(N) time complexity and O(1) memory complexity.

NOTE - the time and memory complexities are calculated in integers. I.e. the input is of size N, not N*logN.

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A ship in the plane has integer coordinates. It also has integer velocity (again in ZxZ). Each turn the ship advances according to its velocity. Here is an example of a ship with velocity (3, 1).

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In this article you will find a collection of riddles. They are all either very well known or extremely easy. Enjoy!

Cutting the Cake

How can you cut a circular cake to eight even pieces with 3 cuts? What is the maximum number of pieces possible with 4 cuts?

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You are given 23 whole numbers, not necessarily distinct, in a row.

You cannot change the order of the numbers.

Prove that there exists an arrangement of the symbols ’+’, ‘×’, ‘(‘ and ‘)’ in-between the 23 numbers, such that the final result is a valid formula, whose evaluated value equals 0 mod 2000.

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 There are 100 seats in an airplane.

There are 99 male passangers with reserved seats and one female passager that does not have a ticket.

The female passanger enters the plane first, selects a random seat (out of the total 100 seats) and sits in it.

Then the first man enters. If his seat is not taken he sits in it. If it is taken he selects a non-occupied random seat in the plane and sits there.

The rest of the men enters and does the same – each of them first tries to take his own seat (if it is available) and otherwise sits in a random non-occupied seat.

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 There are 100 coins on the table. 90 of the coins have their Heads face up, the other 10 have their Tails face up. Your task is to split the coins into two groups, such that both groups contain the same number of coins with their Heads face up. You are even allowed to flip the coins…

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This one is a riddle of my own invention. It gives a very good counter-example to something we tend to take for granted. Do not be discouraged – I posed it to one of my smartest professors and he did not find the answer! (at least not in the first five minutes…).

The riddle requires some knowledge of Topology and Real-Analysis.  For those of you lacking it, all the relevant definitions are included at the end (I recommend skimming through them before reading the riddle itself).

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