If White is to play, can he always mate Black in 2 moves, regardless of the moves played before?

SOLUTION

First, we notice that since Black made the last move, either the king or the rook has been moved, consequently rendering Black unable to castle. Now White plays Qa1 and no matter what is Black’s next move, Qh8 gives check-mate.

Your last ping pong ball falls down into a narrow pipe embedded in concrete one foot deep. How can you get it out undamaged if all you have is your tennis paddle, your shoelaces, keys, wallet, and a plastic water bottle, which does not fit into the pipe?

SOLUTION

Using the plastic bottle, pour water into the pipe so that the ball will rise up.

Four grasshoppers start at the ends of a square in the plane. Every second one of them jumps over another one and lands on its other side at the same distance. Can the grasshoppers after finitely many jumps end up at the vertices of a bigger square?

SOLUTION

The answer is NO. In order to show this, assume they can and consider their reverse movement. Now the grasshoppers start at the vertices of some square, say with unit length sides, and end up at the vertices of a smaller square. Create a lattice in the plane using the starting unit square. It is easy to see that the grasshoppers at all times will land on vertices of this lattice. However, it is easy to see that every square with vertices coinciding with the lattice’s vertices has sides of length at least one. Therefore the assumption is wrong.

They start with 10 nodes on a sheet of paper and, taking turns, connect any two of them which are not already connected with an edge. The first player to make the resulting graph connected loses.

Who will win the game?

Remark: A graph is “connected” if there is a path between any two of its nodes.

SOLUTION

The first player has a winning strategy.

His strategy is with each turn to keep the graph connected, until a single connected component of 6 or 7 nodes is reached. Then, his goal is to make sure the graph ends up with either connected components of 8 and 2 nodes (8-2 split), or connected components of 6 and 4 nodes (6-4 split). In both cases, the two players will have to keep connecting nodes within these components, until one of them is forced to make the graph connected. Since the number of edges in the components is either C^8_2+C^2_2=29, or C^6_2+C^4_2=21, which are both odd numbers, Player 1 will be the winner.

Once a single connected component of 6 or 7 nodes is reached, there are multiple possibilities:

The connected component has 7 nodes and Player 2 connects it to one of the three remaining nodes. Then, Player 1 should connect the remaining two nodes with each other and get an 8-2 split.

The connected component has 7 nodes and Player 2 connects two of the three remaining nodes with each other. Then, Player 1 should connect the large connected component to the last remaining node and get an 8-2 split.

The connected component has 7 nodes and Player 2 makes a connection within it. Then, Player 1 also must connect two nodes within the component. Since the number of edges in a complete graph with seven nodes is C^7_2=21, eventually Player 2 will be forced to make a move of type 1 or 2.

The connected component has 6 nodes and Player 2 connects it to one of the four remaining nodes. Then, Player 1 should make a connection within the connected seven nodes and reduce the game to cases 1 to 3 above.

The connected component has 6 nodes and Player 2 connects two of the four remaining nodes. Then, Player 1 should connect the two remaining nodes with each other. The game is reduced to a 6-2-2 split which eventually will turn into either an 8-2 split, or a 6-4 split. In both cases Player 1 will win, as explained above.

A long time ago there was a kingdom, isolated from the world. There was only one way to and from the kingdom, namely through a long bridge. The king ordered the execution of anyone caught fleeing the kingdom on the bridge and the banishment of anyone caught sneaking into the kingdom.

The bridge was guarded by one person, who was taking a 10-minute break inside his cabin every round hour. Fifteen minutes were needed for a person to cross the bridge and yet, one woman managed to escape the kingdom. How did she do it?

SOLUTION

Once the guard entered the cabin, the woman started crossing the bridge for 9 minutes, and then turned around and pretended to be going in the opposite direction for one more minute. When the guard caught her, she said she was trying to enter the kingdom, so he banished her away.

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