The Exact Chaos Spreadsheet Game

John D. Cook points out the following interesting fact:

Pick a number x between 0 and 1. Then repeatedly replace x with 4x(1-x). For almost all starting values of x, the result exhibits chaos. Two people could play this game with starting values very close together, and eventually their sequences will diverge.

Cook's on to something here. This does have the makings of a simple game that ties in to math and computing concepts like functions, iterations, spreadsheets and chaos. Here are the rules:

The first player picks a to number from 1 to 99 with the only condition being that the number can't have a five in it (with a little trial and error you should be able to figure out why).

The second player can either steal the first player's number or come up with one of his or her own. If the number is stolen, the first player has to come up with another number.

Both numbers are divided by one hundred and put into the first row of a spreadsheet. Under the first type


=4*A1*(1-A1)


Under the second

=4*B1*(1-B1)



Now copy the formulas going down twenty rows. The copy function should produce an iterative function, with =4*A1*(1-A1) followed by =4*A2*(1-A2) and so on.

Have the students make a line graph of the two series. The winner is the one who ends up with the largest number.

Kriegspiel and Dark Chess

I've been talking about games of perfect information and I have another post on the subject coming up so this seems like a good time to mention two of the best known chess variants of imperfect information.

From the nice people at Wikipedia:

Kriegspiel (German for war game) is a chess variant invented by Henry Michael Temple in 1899 and based upon the original Kriegsspiel developed by Georg von Rassewitz in 1812.[1][2] In this game each player can see their own pieces, but not those of their opponent. For this reason, it is necessary to have a third person (or computer) act as a referee, with full information about the progress of the game. When it is a player's turn he will attempt a move, which the referee will declare to be 'legal' or 'illegal'. If the move is illegal, the player tries again; if it is legal, that move stands. Each player is given information about checks and captures. They may also ask the referee if there are any legal captures with a pawn.

and

Dark chess is a chess variant with incomplete information, similar to Kriegspiel. It was invented by Jens Bæk Nielsen and Torben Osted in 1989. A player does not see the entire board, only their own pieces (including pawns), and squares where these pieces could move.

I've never actually played either of these games (I have enough trouble with unvaried chess), but they raise all sorts of interesting questions about forming a strategy with incomplete information.

Mutant sprouts

A while back I posted a recommendation for a popular pencil-and-paper game:

On the subject of topology, my game of choice is Sprouts, invented by mathematicians John Horton Conway and Michael S. Paterson at Cambridge University in 1967 (as a general rule, you can't go wrong with a game if Conway had anything to do with it).

The rules are simple:

1. Start with some dots on the paper. The more dots you have the longer the game takes so you will probably just want to start with two or three.

2. Players take turns either connecting two of the dots with lines or drawing a line that loops back and connects a dot with itself.

3. The lines can be straight or curved but they can’t cross themselves or any other lines.

4. Each dot can have at most three lines connecting it

5. When you draw a line put a new dot in the middle.

6. The first player who can’t draw a line loses.

I was thinking about sprouts the other day and a few variations occurred to me. I don't know if they're particularly playable or if they add any interesting aspects to the game, but if you can't put a half-baked idea in a blog, what's a blog for?

Variant 1 -- Free sprouts

Played as above but with the following addition: for the first k moves of a game with n dots, the player, after drawing a line, adds a new dot.

Topologically the result is a game with n+2k dots (keep k small) but with the complication that lines are being drawn without knowing exactly how those lines partition the surface. This is still a game of perfect information but the variation should make it more difficult to think a few moves ahead.

Variant 2 and 3 -- Scored sprouts

Each player starts with a separate sheet of paper and proceeds to connect the dot according to the standard rules. After no more lines are possible, the players score their graphs based on the number of dots.

Score = 6


Score = 7

In variant 1 the player with the highest score wins. In variant 2, the win goes to the lowest.

Originally posted at Education and Statistics.

The Blue and the Gray

The famous game designer,* Sid Sackson, had over eighteen thousand games in his personal collection so making his short list was quite an accomplishment, particularly for a game that almost nobody has ever heard of.

On this alone, the Blue and the Gray would be worth a look, but the game also has a number of other points to recommend it: it only takes about three minutes to learn (you can find a complete set of rules here); it is, as far as I know, unique; it raises a number of interesting and unusual strategic questions; and for the educators out there, its Turn-of-the-Century** origins provide some opportunities for teaching across the curriculum. My only major complaint is that it requires a dedicated board, but making your own shouldn't take more than a few minutes.

The object of the game is to be the first to get your general to the center by moving along the designated path while using your soldiers to block your opponent's progress. Since soldiers can capture each other, the game has two offensive options (capturing and advancing) compared to one defensive option (blocking). (Something I learned from developing my own game was the more the designer can shift the focus to offense, the better and faster the game will play.)

I don't know of any attempt to do a serious analysis of the Blue and the Gray. Might be fun to look into. If someone out there finds anything interesting, make sure to let us know.


* Yes, I did just use the phrase, 'famous game designer.'

** I'm going off memory here about the age of the game. You should probably double check before building a lesson plan around this. (see update)

From West Coast Stat Views

UPDATE:

Via the good people at the University of Maryland, here's the original patent from 1903.