Alea Iacta Est

So you have found it! Here is the first in what will hopefully be a series of strange (but perfectly playable) games, devised with all of my twisted cunning and general weirdness. In its natural form, it is called Icosawalk, for reasons that may or may not become clear soon.

The game requires 19 players. A couple fewer may be possible, but too far off 19 and you're stuffed. (Some of the variations, below, cater for fewer players, or in extreme cases, more). A model icosahedron would be very handy. (I'll see if I can find a link to somewhere that has instructions for the origami one I can do).

With the faces of your icosahedron carefully numbered (there should be twenty; if not something is seriously wrong), everyone pick a number. No duplicates please, and there should be one left over. Don't worry if you haven't got a model icosahedron yet; I'll give instructions for doing without one later.

The object of the game is to 'walk' from your starting face - the one with the number you picked - to the one on the opposite side of the figure. (Again, if you've got a shape without obvious opposite faces, something is wrong). Play proceeds as follows.

On the first turn, each of the players on sides adjacent (sharing an edge) to the empty one must give a brief speech to the rest of the players, explaining why they should get the first move. A vote is then taken, with all of the players (including those standing for election) casting one vote each (no abstention is allowed). The player with the most votes moves onto the empty face. Special case: In the unlikely event of a tie, the third player, who has fewer than either of the others, moves instead.

In the second and subsequent turns, the player who just left a face may not be voted back onto it, so the vote is between the other two adjacent players. Because no abstention is allowed, ties are no longer possible (unless you're playing with a non-standard number of players, but that isn't my problem).

Note that in later turns a player may be encouraging you to vote against them, because the move would be away from their goal. Consider also if you are playing from a table (rather than a model, see below), it will be quite hard to see who is close to reaching their goal. For this reason you should make a point of being prejudiced against whoever seems to deserve it, and should attempt to get as much real-world gain as possible.

One or two more rules need to be mentioned:

• Any promises or pledges made during the game must be adhered to when the real world is returned to. The other players will be responsible for adjudicating this.
• Anyone who forgets where they are loses immediately, and should be shouted at and otherwise punishèd.

But I have no model icosahedron (sob, sob...)

I have yet to look for modular origami on the web, and I'm not going to tell you how I do it, since I learnt it from a book. However, provided you're playing on faces, you can get hold of an appropriate die (yes, they do come in other shapes than just cubes...).

Note: As a general rule, for a properly numbered die, the sum of opposite faces (i.e. a start face and its target) will be equal to the sum of the highest and lowest numbers on the die (21, for icosahedra).

For those without even a die, here is a table of faces, their opposites (for the goal), and their adjacent:

Variations

There are many potential variations on Icosawalk. Here are just a few.

For fewer players: Play on a cube (5 players) or an octahedron (7 players). These should also be easier to visualise and keep track of, but potentially no less interesting. One thing I don't recommend is playing on a tetrahedron. The 7 player version could be called Octowalk, but people who are or have been role-players, war-gamers or other weird pastime involving strange polyhedra might want to call the 5 player one 'Walk', since clearly everyone else thinks the default thing is to have six sides...

For more players: In theory a table can be drawn up for almost any number of players, just by mapping out the vertices of a polyhedron with one more vertex than players. Various problems appear, with figures with ill-defined 'opposite vertices', for example, but if you're the kind of person who wants to play this thing on anything more dramatic than an icosahedron, then you can probably work something out.
A note about fairness: Play on a Platonic solid is intrinsically fair, since the shape has complete symmetry; the most it can get for its number of sides. Other shapes (ten sided dice, for example), do not have this property, and may benefit certain start positions.

For the interest of Geometer types: Note that other numbers of players can also be gained by playing on vertices and edges rather than faces, but that this is (for the Platonic solids, certainly) equivalent to playing on the faces of a different figure; icosahedron to dodecahedron and back, or cube to octahedron and back.
The faces of a tetrahedron are equivalent (for these purposes) to its vertices. Go and find a real shape already!

For people with a bigger model, or too much spare time: If you have, or have the time to make, a big enough figure of the right kind, you could stick flags on it, or something equally daft, to mark everyone's position. A huge icosahedral piece of Edam would be ideal, combined with 'specially made cocktail-stick flags.

For people in a more accommodating universe: Find a model with sufficient mass, and mark your position by standing on the correct face. There are a few things to note about this approach:

• Do not place your model too close to any celestial bodies. The gravity on your players due to nearby planets etc. may be too great.
• Bear in mind that unless really quite dense, the figure may be large enough to require some kind of powered transport to move people between faces. Certainly telecommunications equipment will be vital. If you had a helicopter in mind, remember that you will then need atmosphere.
• If you are constructing the object yourself, be careful not to skimp on volume too much. In your drive to keep the thing to a manageable size while maintaining the required mass, you may accidentally give it the density required for it to become a black hole.

For people who are insane, or mathematicians: Take away the model, and make everyone use their power of visualising R³.

For people who are insane AND mathematicians: See above, under Fewer Players. See the 'Walk'? On a cube? OK. One word. Tesseract.