# Professor's game

1. Apr 3, 2007

### z-component

This is a strategic game my math professor gave our class. I'm not sure of the different strategies, but this may serve as a fun "brain teaser" for you all. Also, if anyone knows that this is a previously-made game, please provide the name of it.

View the board:

The objective:

There are two players: the circle player and the pursuing player. The goal of the pursuing player is to bound (trap) the circle player (more on this below) and the goal of the circle player is to escape being trapped. I'm not sure if there is always a definite winner.

The rules:

I had a rough 5 minute intro to the game so these rules are what I've gathered. Again, if anyone knows any official rules to the game, please provide them.

The two players alternate making moves in order to accomplish their goal. The circle player may move in any direction (horizontally, vertically, diagonally) on the gridded board, one box at a time. The pursuing player must trap the circle by shading in boxes around the circle (one box per turn). However, the circle may "jump" over a single box (kind of like checkers) but not more than one. Therefore, if the circle is completely surrounded by boxes with more than one "layer," he is trapped.

This game is similar to chess, where there is the possibility of "check" and "checkmate." I.e., the circle is checked if he is surrounded by a single box on all sides, but may avoid this threat by jumping. Checkmate is assumed when he is trapped and has no way to jump or move outside of the boxes.

I hope this is clear!

2. Apr 3, 2007

### neurocomp2003

Um there questions:
[0] is there a step/move limit for the pursuing player
[1a] is there any rule to handle if the circle guy and square guy are on the same square?
[1b] if 1a allows for same square, can circle guy jump over a black square even though he is on one himself?
[2] can circle jump a horses manouvre in chess?

3. Apr 3, 2007

### Stainsor

fun :tongue2:

4. Apr 3, 2007

### z-component

0-There is no limit to the number of moves each player has.
1a-If you shade a box that the circle is on, the circle can just move to another box.
1b-I'm assuming yes.
2-No. Only one box at a time.

5. Apr 4, 2007

### davee123

Yeah, I would have to say that you need a couple other stipulations:

1) Turn limit and/or scoring

If you have a 10x10 grid (say), there's an obvious limit to how long the game can continue. Clearly, the square player can only go 100 times, after which the circle player is necessarily trapped. In other words, if there is no turn limit, the circle player will ALWAYS lose eventually. Therefore, some possibilities:

A. Impose a turn limit of (say) 50 turns (or however many turns is half the number of grid squares).

B. Keep score. Each player plays once as the circle player, and sees how long they can survive. Whichever player survived for longer when they were the circle is declared the winner.

C. Allow victory conditions for the circle, such as "visit all four edges of the board" or "jump back and forth between the same two squares 3 times in a row".

2) Invalid moves

A. "The circle player may not move onto a square that has already been shaded." I assume that you meant to say this, but I'm not sure. If this isn't a rule, the game is made much easier for the circle player.

B. "The circle player may not jump over blank squares, he may only jump over shaded squares." Again, I sort of inferred this rule from what you said, but you didn't actually say it, so I dunno.

C. "The circle player may not exit the board." I'm just assuming this to be true, and assuming that nothing funky happens like the circle exiting one side of the board and re-appearing on the other side.

DaveE

6. Apr 4, 2007

### neurocomp2003

the turn limit will probably be #sqaures-1

7. Apr 4, 2007

### davee123

Nah, that wouldn't work. If the there are N squares, and the square player decided to fill the corners in last, then he would ALWAYS be guaranteed a win on the N-4th move. Actually, it's trivially easy to show that the square player can be guaranteed victory on about the 8N/9th turn. Just fill in the board something like this:

Code (Text):
+--+--+--+--+--+--+--+--+--+--+
|  |XX|XX|  |XX|XX|  |XX|XX|  |
+--+--+--+--+--+--+--+--+--+--+
|XX|XX|XX|XX|XX|XX|XX|XX|XX|XX|
+--+--+--+--+--+--+--+--+--+--+
|XX|XX|XX|XX|XX|XX|XX|XX|XX|XX|
+--+--+--+--+--+--+--+--+--+--+
|  |XX|XX|  |XX|XX|  |XX|XX|  |
+--+--+--+--+--+--+--+--+--+--+
|XX|XX|XX|XX|XX|XX|XX|XX|XX|XX|
+--+--+--+--+--+--+--+--+--+--+
|XX|XX|XX|XX|XX|XX|XX|XX|XX|XX|
+--+--+--+--+--+--+--+--+--+--+
|  |XX|XX|  |XX|XX|  |XX|XX|  |
+--+--+--+--+--+--+--+--+--+--+
|XX|XX|XX|XX|XX|XX|XX|XX|XX|XX|
+--+--+--+--+--+--+--+--+--+--+
|XX|XX|XX|XX|XX|XX|XX|XX|XX|XX|
+--+--+--+--+--+--+--+--+--+--+
|  |XX|XX|  |XX|XX|  |XX|XX|  |
+--+--+--+--+--+--+--+--+--+--+
If you're a clever square player, I'm sure you could do even better. There's likely a much more optimal arrangement of filled in squares from which the circle has no escape. Yeah, I can get a more optimal solution that's victory at about the 4N/5th turn, by arranging the empty squares like knights in chess.[/edit]

Basically, if my assumptions about the rules are correct, the odds are GREATLY in favor of the square player. But by imposing a turn limit (or other win condition for the circle player), you can even the odds not to favor the square player so much.

Pft. Ok, obviously I was wrong about some of my assumptions. The circle player HAS to be allowed to go onto filled-in squares, because otherwise, on a 100x100 board, the square player can guarantee victory in 39 turns.[/edit]

DaveE

Last edited: Apr 4, 2007