1 2 3 Foot

1 2 3 Foot is a great game.  Generally, it is played when there is a group of people who must determine a subset of that group who must do a specific thing.  For example, if there were five dudes hanging out and they decided to go to McDonald's and they wanted one guy to drive everyone, they would play 1 2 3 Foot to determine who drives.  This guy is the winner, regardless of whether he wanted to drive or not.  There does not have to be one winner though, there can be more - you might use it to determine which two guys have to carry a heavy couch or which three guys get to eat the last three pieces of pizza.  You can play the game to determine who must do something that is no fun or who gets to do something that is fun.

how the game works:
First, specify how many winners there will be.  Then, all players stand around a circle and chant, "1 2 3 Foot," while alternating between putting their bad and good foot forward in unison with the words they chant.  By this, I mean when they say, "1," they put their bad foot forward, when they say "2," they put their good foot forward, bad foot again for "3", but on "foot", they put their good foot forward in one of four ways - they elevate their heel while keeping their toes down (known as "down"), they elevate their toes while keeping their heel down (known as "up"), they rotate their foot 90 degrees clockwise ("in" if you are a lefty, "out" if you are a righty), or they rotate their foot 90 degrees counter-clockwise ("out" if you are a lefty, "in" if you are a righty).  Next the players compare what they have each done.  If their is exactly one group of people that did the same thing with their feet that is the same size as the number of winners specified, this group is the winner(s).  That last sentence may have been confusing, so here's an example.  Six players assemble to determine two winners.  One guy puts his foot out, no one puts their feet in, two guys put their feet up, and three guys put their feet down.  These two guys are the winners.  Many times, the match will result in what is known as a "foot".  A foot occurs when no winner is determined.  Consider the previous example.  If instead, one guy puts his foot out, one guy puts his foot in, two guys put their feet up, and two guys put their feet down, their were two sets of two.  These guys tied for the win, so that round was a foot.  A foot also occurs when their are no groups of the correct size.  Consider the 6 player and two winner scenario once more.  If three people put their feet up, three people put their feet down, and no one put their feet in or out, there are no groups of size two.  They have achieved a foot.  Achieving a foot may be more good than bad, though.  When a foot occurs, players get to play another round.  What a treat!  This is where things get complicated, though.  Each time there is another round, players switch between counting to three and counting to four.  So if there is a foot in round one, play another round and count to four.  Players must remember to stick their good foot forward on "1", so that their good foot will be forward on "foot".

how often do foots occur:
I made an R function which simulates 1 2 3 Foot games to find probabilities of achieving a foot.
Here's the code:

one23footFootProb=function(players,sim,winners=1){
foot=0;
for(i in 1:sim){
vec=sample(1:4,players,replace=T);
vec2=numeric(4);
for(j in 1:4){
vec2[j]=sum(vec==j)}
if(!sum(vec2==winners)==1){foot=foot+1}}
return(foot/sim)
}
"players" is the number of players, "sim" is the number of simulations and "winners" is the number of winners. The function returns the proportion of times that a foot occurred in the simulation.  Here's an example:

> one23footFootProb(5,10000,2)
[1] 0.6439
In a 5-player match to determine 2 winners, there was a foot in 64.39% of the rounds.  This means that the average match lasted 1.553036 rounds.

> 1/.6439
[1] 1.553036

Here are 10000 trials for matches between one and thirty players to determine one winner.

> foots=numeric(30)
> for(i in 1:30){
+ foots[i]=one23footFootProb(i,10000)}
> foots
 [1] 0.0000 1.0000 0.4417 0.8231 0.5908 0.6310 0.5862 0.5377 0.4925 0.5023
[11] 0.5363 0.5924 0.6364 0.7012 0.7447 0.8009 0.8348 0.8686 0.8986 0.9182
[21] 0.9378 0.9494 0.9587 0.9693 0.9753 0.9826 0.9849 0.9869 0.9882 0.9938
For the most part, the odds of a foot increases as the players increase.  What's interesting is how often foots occur in four player matches.  That's because a foot occurs only when three people do one thing and one person does another.

Note that this simulation assumes that each player will do each foot move with equal probability in each round.  This is not a correct assumption.