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Previous problems and their solutions.

Problem 1: A sweet game. Solution.

This is a very famous problem - yet a good place to start!
Two players take turns in breaking a chocolate bar of 6x8 cells into smaller pieces. In each move a player picks one of the pieces from the table, breaks it into two pieces strictly along one of the straight lines between the cells and puts the pieces back on the table. The player that cannot make his move looses the game. Find out which player has a winning strategy, the 1st or the 2nd one.

Problem 2. Solution.

It is well-known that a whole number is divisible by 3 if and only if the sum of its digits is divisible by 3.
It seems far less known why this is so. If you do understand why, it should not be hard for you to do the following:

Find a criterion of divisibility by 11.

Problem 3: Caribbean Pirates. Solution.

Three pirates share loot (consisting of various coins, pearls, noble metals, precious stones, rings and other jewelry). Each of the pirates is sure that he can divide loot fairly, but the other two don't trust him. If there were only two pirates, the first one would divide the loot, and the second one would choose the half which he thinks is bigger. What should the three pirates do so that in the end each of them is sure he got at least one third of the loot?

Problem 4. Solution.

Problem 5. Solution.

My daughter Alisa claims that in every class at Tehiyah there are two students who have the same number of friends within the class. Is she right?

Problem 6. Solution.

Twenty points are placed around a circle. Two players take turns joining two of the points with a straight line segment which does not cross or meet a segment already drawn in. The player who cannot do so loses. Which player has a winning strategy, the 1st or the 2nd one?

Problem 7. Solution.

Convert 10/17 into a decimal fraction.

Problem 8. Solutions.

The map of United Flatland consists of several straight lines dividing the infinite plane into states . Show that the map can be painted into two colors in such a way that no two states which share a segment of border have the same color.

Problem 9. Solution.

On his hike from town A to town B James Bond has to meet an assistant driving along the straight road CD. Help James to choose the meeting point X that would minimize his hiking distance |AX|+|XB|.

Problem 10. Solution.

Find the greatest common divisor of 50399 and 128243.

Problem 11. Solution.

Are all numbers of the form n^2+n+41 prime?

Problem 12. Solution.

Two white and two black knights are positioned on the 3x3 chessboard as shown on the left picture. Can they move, using the usual chess knight's moves, to the position shown on the right picture?
Picture for Problem 12