### Ask the Chicken Philosopher

*Dear Olivia,*

*I am a chicken farmer with 3 Rhode Island Reds and 3 white leghorns. Last week I went to close up my little coop for the evening and noticed all the chickens were sitting in a line with the 3 whites on one side and the 3 reds on the other. Normally they just arrange themselves randomly, without any apparent preference for who they roost next to. I was so surprised that I ran into the house to get my camera, but by the time I got back it was so dark that I couldn’t take a decent photo.*

*This got me wondering, Olivia, how long might I expect to wait to see this remarkable sight again. What about for other interesting arrangements?*

*Sincerely,*

* Chicks on Sticks*

* Portland, Oregon*

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Dear Chicks,

You don’t say how long the roosting stick in your ‘little coop’ is and this turns out to be a very important piece of information.

If there is only space for 6 chickens, we can think of the roost as a series of 6 spaces, where each space can hold one chicken. The question we need to answer is: “How many possible ways are there to arrange 6 chickens in 6 spaces?”

Once we place the red chickens, we will have also chosen where the white ones go as well. So, the question is really: How many ways can we choose 3 spaces from 6 spaces?

It turns out that there are only 20 ways, found from the binomial equation:

where *S* is the number of spaces and *R* is the number of Rhode Island Reds. The exclamation point (!) indicates a factorial. That is, 3! is 3 x 2 x 1 and 6! is 6 x 5 x 4 x 3 x 2 x 1.

(These numbers are easy to calculate if *S* and *R* are small, but you wind up doing a lot of unnecessary multiplication for larger values.)

Of these 20 configurations, only 2 of them have all three reds on the left or all three on the right. So, if your chickens arrange themselves randomly, you should see them in this arrangement about 10% of the time, or about once every week-and-a-half.

However, if the roosting stick is longer than 6 spaces, the number of possible configurations of chickens goes up dramatically. For example, if the stick is 9 spaces long, then we must choose 3 spaces from 9 in which to first place the Reds. Then, we must choose 3 spaces from the remaining 6 in which to place the Whites.

The number of ways to place 3 Reds into 9 Spaces is given by the equation above and is equal to 84. For each of those 84 ways there are, of course, 20 ways to place 3 Whites in the 6 remaining Spaces.

Therefore, when empty spaces are present, the number of possible arrangements of *R* red chickens and *W* white chickens into *S* spaces (with *E* empty spaces) is:

In statistical mechanics, this is called a ‘partition function’. Since each arrangement of chickens is equally likely, it simply tells us the total number of possible arrangements.

For this particular case, the number of arrangements is 1680. Of these, 20 have 3 reds in a row and 3 whites in a row, with no spaces between the respective blocks of reds or whites. So, if the roosting stick is 9 spaces long, you should only see one of these configurations about 1% of the time, or about 3 times per year. Far fewer than the previous case.

Of course, being a chicken myself, I can tell you that we don’t like to spread ourselves randomly on a roosting stick when we can huddle together instead. Regardless of the length of a stick, if there are 6 of us, we’ll probably treat it as if it’s only 6 spaces long.

Love,

Olivia

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*Olivia DeLane has a Master’s certificate in normative ethology from Gallus College, a non-accredited online institution. Her writings are for entertainment purposes only and should not be misconstrued as being for any other use. Olivia’s screenplay “Shell Game”, an intense action-packed thriller set on a commercial chicken ranch, is currently being made into a major Hollywood blockbuster starring Gary Oldman as a nefarious egg farmer clinging to the edge of sanity and Brad Pitt as ‘the chicken who knows too much’.*

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