Hanoi Rules

An abstract strategy game
of planning and scheduling
for two or three players.

Pieces: Two players, White and Blue, each have 15 discs of their colour, three each of the following sizes.

Start: The game is played on a size 3 hex hex board and starts as shown. The owner of the topmost disc owns each stack.

Aim: The aim is to complete a multicoloured tower of height 5 consisting mostly of your discs.

Play: White starts, then players alternate taking turns. Each turn the mover must choose a stack of theirs to split.

Split: The stack is split by moving each disc in turn, from top to bottom, regardless of colour, in a straight line of the mover's choice. Different discs can move in different directions.

Each disc must move along its line to land on the closest disc of size N+1, if any, else the furthest empty cell. Discs move over intervening stacks unless they're size N+1.

The following example shows a typical split move by White (left) and reply by Blue (right). Note that Blue's 1-disc and 4-disc land on the White's 2-disc and 5-disc respecitvely; these white discs are now pinned and under Blue's control.

Summary: Move along lines to the closest N+1 disc, else the furthest empty space.

End: The game ends when a multicoloured stack of height 5 is formed, and is won by the player who owns the majority of discs making up this stack. For example, Blue wins the following game 3:2.

Notes

A single disc counts as a stack of height 1.

When splitting a stack, each disc must move if there's a legal move available. A player's turn stops if the current (topmost) disc has no legal moves. However, it's not necessary to move the maximum number of discs per turn, i.e. if move A deploys four discs from a stack but move B only deploys three discs from the same stack, then the player can make either move.

It's not mandatory to make a pinning move if other lines of movement are available. That is, each disc deployed during a split must pin the closest N+1 disc if such a disc exists along its chosen line, but the player is free to choose another (non-pinning) line if they want.

Discs are marked with grooves to make their size more obvious. If you stack a disc and can see grooves on the disc below, then you are violating the "N on N+1" rule. The following example shows an illegal 2-on-4 stacking (left) and two valid stackings of 3-on-4 and 2-on-3.

The "N on N+1" constraint means that stacks only grow by consecutive (decreasing) size, i.e. no stack will skip a size and all will have the same "lean angle".

To remember the movement rules, it can help to think of each disc as trapping any N-1 disc that passes over it. The exact movement algorithm is as follows:

    for each disc D in the stack, from top to bottom
        if no possible moves
            end turn
        else
            choose a playable line L
            if exposed N+1 disc along L
                move D onto the closest exposed N+1 disc along L
            else
                move D to the furthest empty cell along L

Tactics and Strategy

Larger discs have the capacity to deploy more troops in a single move. Smaller discs are more mobile and harder to block.

To impede a disc of size N, move a disc of size N+1 in its way. The blocker can be of the opponent's colour.

Like the Tower of Hanoi puzzle, it's generally necessary to split stacks in order to create better ones. Games typically involve cycles of deployment and reassembly.

Three-Player Version

The three-player version of Hanoi starts from the following position and uses the same rules, with the following additions:
• Players must pass if they have no legal moves on their turn, but must move if possible.
• In the event of a tied 2:2:1 majority in the final stack, the game is awarded to the minority player.

History

The game was inspired by the famous Tower of Hanoi puzzle. To be honest, the name was as well.

Thanks to Néstor Romeral Andrés for billiards, play testing and suggestions including the split-and-scatter movement rule and the "N on N+1" constraint.

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