Sunome Variations

The main challenge of a Sunome puzzle is drawing a maze. Numbers surrounding the outside of the maze border give an indication of how the maze is to be constructed. To solve the puzzle you must draw all the walls where they belong and then draw a path from the Start square to the End square.

The walls of the maze are to be drawn on the dotted lines inside the border. A single wall exists either between 2 nodes or a node and the border. The numbers on the top and left of the border tell you how many walls exist on the corresponding lines inside the grid. The numbers on the right and bottom of the border tell you how many walls exist in the corresponding rows and columns. In addition, the following must be true:

  • Each puzzle has a unique solution.
  • There is only 1 maze path to the End square.
  • Every Node must have a wall touching it.
  • Walls must trace back to a border.
  • If the Start and End squares are adjacent to each other, a wall must separate them.
  • Start squares may be open on all sides, while End squares must be closed on 3 sides.
  • You cannot completely close off any region of the grid.

In addition, these variations of Sunome have the following extra features:

  • Paths (borders with a hole in the middle) designate places where the solution should pass through.
  • Pits (black squares) designate places where the solution does not pass through.
  • Portals (circled letters) designate places where the solution should pass through and teleport from one portal to the other.
  • Sunome Cubed is solved similarly but on the surface of a cube. The numbers on the top right, top left, and center left of the border tell you how many walls exist on the corresponding pairs of lines inside the grid. The numbers on the center right, bottom right, and bottom left of the border tell you how many walls exist in the corresponding pairs of rows/columns.

Examine the first example, then solve the other three puzzles.

The solutions are shown below.

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.

Escaping the Kingdom

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?

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.

Out of Time

In the position below, Black played a move, but right before he pressed the clock, he ran out of time. However, the judge declared a draw instead of awarding a victory to the opponent. Why?

The rules of FIDE state that if a player runs out of time, their opponent wins the game IF they have a path to victory. If there is no sufficient material, e.g. a King and a Knight against a King, then the game is declared a draw.

In this position, Black played Rxg6 which forces the moves:

  1. … Rxg6+
  2. Nxg6+ Rxg6+
  3. Kxg6+ Qxg6+
  4. Kxg6

This leaves White with a King and a Knight against Black’s King. Thus, White did not have a path to victory and the game was declared a draw.

Unconscious and Bleeding

A man is found unconscious in front of a store at two in the morning. His head is bleeding and there is a brick laying next to him. When the police arrive, they carry the man to jail. Why did they arrest him?

The man was a burglar who tried to break the store’s glass with the brick. The glass turned out to be bullet proof, so the brick bounced back and hit him in the head, knocking him out.