# Pyraos Rules

## Board and spheres

The board consists of 16 holes in a 4x4 arrangement, in which spheres
may be places. There are 30 spheres, 15 black and 15 white. The 30
spheres are exactly enough to build a pyramid with 16 spheres directly
on the board, 9 spheres on the next level, 4 spheres on the third
level, and 1 sphere on the top.
## Object of the Game

The object of the game is to place a sphere at the topmost position of
the pyramid.
## Starting the game

The game is started with an empty board. The players decide who is to
make the first move, and who plays which color. Players take turns
alternately.
## Moves

A move consists of one of the following four operations. The player
making the move may only manipulate spheres of his own color.
### Placing a sphere in the pyramid

A player may place a sphere of his own color in the pyramid on any
unoccupied position which supports a sphere. That is, either directly
on the board, or on top of a square of spheres already in the pyramid.
### Lifting a sphere in the pyramid

If a sphere in the pyramid is not blocked by other spheres above it,
the sphere may be lifted and placed on a position at a higher level.
### Removing spheres from the pyramid

If the placement of a sphere makes a square of the player's color, the
player may remove at most two spheres from the board after placing the
sphere. A square are four spheres side by side.
Removing spheres is possible also when a sphere is lifted. Also
if the placement of a sphere makes several (intersecting) squares, at
most two spheres may be removed.

#### Children's version

To simplify the rules, one may decide that the removal of spheres from
the pyramid is never allowed.
#### Alternate version

### Pass

A player that is not able to make any other move (that is, is out of
spheres, and cannot lift a sphere), may pass the turn to the other
player. A pass is not allowed under any other circumstances.
## Comments

Since there are only just enough spheres to build the pyramid, the
lift and remove moves are of course essential to the game.
When playing the game, it seems that the player that starts has a
slight disadvantage. The intuition is correct, and in fact, at least
when using the standard version of the rules, the second player to
move is guaranteed to win the game if he plays correctly. The
situation for the alternate version of the rules has not yet been
evaluated. In practise, however, human players seem unable to exploit
this game theoretical result.

Staffan Ulfberg