## The Connect Game

Two friends are playing the following game:

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.

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:

1. 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.
2. 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.
3. 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.
4. 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.
5. 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.

## Napoleon and the Policemen

Napoleon has landed on a deserted planet with only two policemen on it. He is traveling around the planet, painting a red line as he goes. When Napoleon creates a loop with red paint, the smaller of the two encompassed areas is claimed by him. The policemen are trying to restrict the land Napoleon claims as much as possible. If they encounter him, they arrest him and take him away. Can you prove that the police have a strategy to stop Napoleon from claiming more than 25% of the planet’s surface?

We assume that Napoleon and the police are moving at the same speed, making decisions in real time, and fully aware of everyone’s locations.

First, we choose an axis, so that Napoleon and the two policemen lie on a single parallel. Then, the strategy of the two policemen is to move with the same speed as Napoleon, keeping identical latitudes as his at all times, and squeezing him along the parallel between them.

In order to claim 25% of the planet’s surface, Napoleon must travel at least 90°+90°=180° in total along the magnitudes. Therefore, during this time the policemen would travel 180° along the magnitudes each and catch him.

There are 7 loaves of bread that need to be shared equally among 12 people. How would you do this if you are not allowed to split any loaf into 12 pieces?

Split 4 of the loaves into 3 pieces and the other 3 loaves into 4 pieces. Then, give each person one of both types of pieces.

## Death Cult

A thousand people stand in a circle in order from 1 to 1000. Number 1 has a sword. He kills the next person (Number 2) and gives the sword to the next living person (Number 3). All people keep doing the same until only one person remains. Which number survives?

First, we note that if the number of people is a power of 2, then the first person will survive every round. The greatest power of 2 that is less than 1000 is 512. Therefore, after 488 people die, there will be 512 remaining and the first one to kill the 489-th person will survive. This person has number 1+2×488=977.

## A String Around a Rod

A string is wound around a circular rod with circumference 10 cm and length 30 cm. If the string goes around the rod exactly 4 times, what is its length?

Imagine the circular rod is actually a paper roll and the string is embedded inside the paper. When we unroll it, we get a paper rectangle 30cm×40cm with the string embedded along the diagonal. Using the Pythagorean theorem, we find that the length of the string is 50cm.

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## Lost In the Forest

You are lost in the middle of a forest, and you know there is a straight road exactly 1 km away from you, but not in which direction. Can you find a path of distance less than 640 m which will guarantee you to find the road?

Imagine there is a circle with a radius of 100 m around you, and you are at its center O. Let the tangent to the circle directly ahead of you be t. Then, follow the path:

1. Turn left 30 degrees and keep walking until you reach the tangent t at point A for a total of 100×2√3/3 meters, which is less than 115.5 meters.
2. Turn left 120 degrees and keep walking along the tangent to the circle until you reach the circle at point B for a total of 100×√3/3 which is less than 58 meters.
3. Keep walking around the circle along an arc of 210 degrees until you reach point C for a total of 100×7π/6 which is less than 366.5 meters.
4. Keep walking straight for 100 meters until you reach point D on the tangent t.

## An Ant’s Path

An ant is positioned at one of the vertices of a cube and wants to get to the opposite vertex. If the edges of the die have length 1, what is the shortest distance the ant needs to travel?

We unfold a cube to get a cross-shaped figure. Then, the problem is to find the shortest path between two points separated by a horizontal distance of 2 units and a vertical distance of 1 unit.

It is easy to see that the path in question is the one passing through the middle of the edge between the start and end points, and which has a distance of √5.

## A Square and an Invisible Point

There is a square drawn on a piece of paper and also a point marked with invisible ink. You are allowed to draw 3 lines on the paper and for each of them you will be told whether the point is on its left, on its right, or lies on the line. Your task is to find out whether the point is inside the square, outside the square, or on its boundary. How do you do it?

Draw one of the diagonals of the square. Then, draw the 2 lines containing the sides of the square that are on the same side as the invisible point.

## Toggle a Pixel

Toggle one pixel to make this equality correct.

(71+1)×(71-1) = 7×6×…×2×1 = 7!

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## Eye of the Dragon

Divide the circle below in two pieces. Then, put the pieces together to get a circle with a dragon, such that the dragon’s eye is at the center of the new circle.

The solution is shown below.