A Flock of Chickens can't have exactly Two Kings

[jdall...@yahoo.com]

I stumbled on a domain of mathematics which is
about as simple as imaginable. It might seem like graph
theory but is much too simple even to be called that.
It might be a good toy domain to kindle youngsters' interest.

Definition: A Flock is a finite non-null set of chickens.
(Think of 'chicken' as an abstract object; never mind
whether the following apply to real chickens or their pecking.)

Property of 'Peck': For any chickens x, y in a Flock,
exactly one of the following is true:
(a) x Pecks y
(b) y Pecks x
(c) x = y
Remark: Pecking need not be transitive. If x Pecks y
and y Pecks z, it may be the case that z Pecks x.

Definition: K is an Emperor of Flock F if and only if
K∈F and
K Pecks every chicken in the Flock except K himself.

Definition: K is a King of Flock F if and only if
K∈F and
for every z∈F who Pecks K, there is some
y∈F with K Pecks y and y Pecks z.

Theorem 1: Every Flock has at least one King.

Theorem 2: A Flock has an Emperor if and only if it has exactly one King.

Theorem 3: ---- [Fill in the blank -- a "secret lemma"!]

Theorem 4: No Flock has exactly two Kings.


Theorems 1 and 2 are easy. The challenge is to prove Theorem 4.
What makes this interesting is that there is a Theorem 3 which is
easier to prove than Theorem 4, and for which Theorem 4 is an easy
corollary. Can you (or a youngster) guess the secret Theorem 3
which assists this proof?

(Or -- also a fun challenge -- prove Theorem 4 without discovering
and applying the "secret lemma.")

James D. Allen

[Frank J. Lhota]

I take it this was prompted by the 2013 LVAIC Mathematics competition.
https://sites.lafayette.edu/liebnerj/files/2014/10/LVAIC2013.pdf

--
"All things extant in this world,
Gods of Heaven, gods of Earth,
Let everything be as it should be;
Thus shall it be!"
- Magical chant from "Magical Shopping Arcade Abenobashi"

"Drizzle, Drazzle, Drozzle, Drome,
Time for this one to come home!"
- Mr. Wizard from "Tooter Turtle"

[James D. Allen]

There are various on-line references to Pecking Order & Kings & Emperors,
on the 'Net and elsewhere. Your cite asks for a proof of Theorem 1.
This is easy by induction: Just show that if K is a King of Flock F, then
either K or T will be a King of (F union {T}).

More interesting is Theorem 4. What makes this "fun" is that, as I wrote,

> > The challenge is to prove Theorem 4.

> > What makes this interesting is that there is a Theorem 3 which is much
> > easier to prove than Theorem 4, but for which Theorem 4 is an easy
> > corollary. Can you guess the secret Theorem 3
> > which assists this proof?

jamesdowallen at Gmail