The battlefield may be any shape, but the standard game is on a triangular 10-board.

One player writes a black and red positive integer on two hexagons. The other player gets to choose to play black or red.

Black always moves first.

Each player gets three life which are the x markers flanking the game board. We will discuss them in a while, but first lets talk about horsemanship and archery.

Horsemanship

At the start of each turn you must ride 1-3 hexagons from your current location. Write the next highest integer in one of these hexagons. In this case the black player will write "51" somewhere in the green area.

Archery

Next it is time to shoot.  You may shoot any hexagon on the board.

Black's turn is finished. It is now time for red's turn and to explain how you can lose.

Horsemanship

Red will write "31" in one of the green hexagons.

Losing the battle

If it were the end of Red's turn, Red would lose because...

Red's current location can see an enemy hexagon that does not share a common prime factor with itself. "50" and "31" do not share a common prime factor.

However, luckily for Red, it is not the end of Red's turn...

Red can still shoot an arrow.

Now it is the end of Red's turn, and thankfully red can no longer see the "50." It is only the closest number or destroyed hexagon that needs to be checked in each direction.

Black plays "52" - is that dangerous for black?

No - it is not dangerous. The "52" can see the "30" without danger because they share a common prime factor.

Black shoots an arrow.

Red makes a risky move and loses one of her three lives. Let's see why it is risky...

The move is risky because Red can see one of her own integers that does not share a common factor with her current number.  "32" does not share a common factor with "31."

This check is made before shooting an arrow. If a risky move is made - the player loses a life and may not shoot an arrow that turn.

Red loses a life. If red loses all three, she has lost the game.

Is black's move risky?

Yes - Black's move was risky: "51" and "53" do not share a common prime factor. Black loses a life and does not get to shoot this turn.

Red's "33" does not see any of her own integers which are relatively prime. Actually she doesn't even see the "30" because a hexagon in-between has been destroyed.

Therefore - Red may shoot an arrow.

That is lucky, because if Red "33" saw Black "50" - Red would immediately lose because the two do not share a common prime factor.

Red saves the game by blocking the line-of-sight between her "33" and Black's "50."

Red does not have to worry about the 51. Why? Because both share a common prime factor of 3.

Black moves to "54."

Black shoots.

Red moves.

Red shoots.

Black makes a risky move...

Black loses a life and does not get to shoot this turn.

Red makes a risky move...

Red loses a life, but now it is Black's turn. Black has nowhere to go. Black loses.

The game board is the first page in this pdf file.