Pointless games and internet toys to waste your time. Not all the ones that used to be on this website are here yet, mainly because some of the older scripts are broken.
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Feel free to click and drag...
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:
| Face | Opposite | Adjacents |
|---|
| 1 | 20 | 7 | 13 | 19 |
| 2 | 19 | 12 | 18 | 20 |
| 3 | 18 | 16 | 17 | 19 |
| 4 | 17 | 11 | 14 | 18 |
| 5 | 16 | 13 | 15 | 18 |
| 6 | 15 | 9 | 14 | 16 |
| 7 | 14 | 1 | 15 | 17 |
| 8 | 13 | 10 | 16 | 20 |
| 9 | 12 | 6 | 11 | 19 |
| 10 | 11 | 8 | 12 | 17 |
| 11 | 10 | 4 | 9 | 13 |
| 12 | 9 | 2 | 10 | 15 |
| 13 | 8 | 1 | 5 | 11 |
| 14 | 7 | 4 | 6 | 20 |
| 15 | 6 | 5 | 7 | 12 |
| 16 | 5 | 3 | 6 | 8 |
| 17 | 4 | 3 | 7 | 10 |
| 18 | 3 | 2 | 4 | 5 |
| 19 | 2 | 1 | 3 | 9 |
| 20 | 1 | 2 | 8 | 14 |
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.
Oh, so you followed the link? Hmm. I don't know if you'll like it in here. Anyway, I would have
a model hypercube for you, but the applet I'm using can only handle three dimensions. Odd that.
If I ever get a good enough understanding of how it works, I may extend it to four, but that's some
way off yet.
Oh, and in case you don't know, would anyone like me to explain tesseracts
to them?
So without further ado, let me tell you about Tesserawalk. Firstly, it's not as simple as before.
Before we used faces, and the word faces was well defined. Now it is less so. If anyone works out
a good way of using the cubic faces of the polytope itself, or the faces of the faces, I would
welcome suggestions. Until then, I'm using the vertices, and their connecting edges.
Just out of interest, how many right angles do you think you can see in the
diagram above? I mean the picture itself, rather than the actual figure (since
all of the angles in the hypercube are right).
There are 16 vertices, with 4 connecting edges each. This gives fewer players than Icosawalk,
but with more candidates per vote. One version of the table is as follows:
| Vertex | Opposite | Adjacents |
|---|
| 1 | 16 | 6 | 9 | 13 | 15 |
| 2 | 15 | 5 | 10 | 14 | 16 |
| 3 | 14 | 8 | 11 | 13 | 15 |
| 4 | 13 | 7 | 12 | 14 | 16 |
| 5 | 12 | 2 | 9 | 11 | 13 |
| 6 | 11 | 1 | 10 | 12 | 14 |
| 7 | 10 | 4 | 9 | 11 | 15 |
| 8 | 9 | 3 | 10 | 12 | 16 |
| 9 | 8 | 1 | 5 | 7 | 14 |
| 10 | 7 | 2 | 6 | 8 | 13 |
| 11 | 6 | 3 | 5 | 7 | 16 |
| 12 | 5 | 4 | 6 | 8 | 15 |
| 13 | 4 | 1 | 3 | 5 | 10 |
| 14 | 3 | 2 | 4 | 6 | 9 |
| 15 | 2 | 1 | 3 | 7 | 12 |
| 16 | 1 | 2 | 4 | 8 | 11 |
A Hypercube (or Tesseract, or Measure Polytope) is a cube extended into four
dimensions. It has sixteen vertices, 28 edges, eight cubic 'hyperfaces' and
goodness-knows-how-many square faces (count them if you like; I can't be
bothered).
If you don't believe that there are eight faces (it is quite hard to
visualise, and it took me some time to find them all), go back up to the top,
and look at the image up there. When you roll the cursor over the writing to the
right, it should highlight the different faces. Bear in mind that most of them
don't look cubic; that's a two dimensional representation of a four dimensional
figure, so it's bound to look funny.
It is generated by, well, let's assume for a moment that we don't know what a
cube is. In no dimensions, the only possible figure is a point. Get hold of an
extra dimension from somewhere (don't ask where; if I knew that I would be selling
the knowledge of how to walk 'through' walls right about now). Extend your point
through a finite distance in that dimension.
It might be helpful if your distance was one unit, but I don't mind. If you're
trying to be awkward, I'll just define a new unit equal to the length of your line segment.
So anyway, now you have a unit [one unit long] line segment. Get another dimension,
perpendicular to the one you already have.
Extending the line through a unit in this new direction gives a square. Get another
dimension, perpendicular to both of the others. Extending in this new one gives a cube.
Notice the way that on the diagram, the third dimension doesn't seem perpendicular. That's
because you only have a two-dimensional display, and if I were to use a mutually orthogonal
[perpendicular] direction it would have to go into or come out of your screen.
Now we have a problem. There isn't a dimension perpendicular to all three of the ones we have.
Not in our space at least. But mathematicians care not for such problems, so suppose
there is such a direction, and extend through one unit in it.
We now have an object that can't be drawn, can't be built, and is of no use to
us whatsoever. Except perhaps for playing Tesserawalk...
