# Infinite Defenestration

Infinite Defenestration is a method of proof which utilizes repeatedly throwing members of a party out a window. It relies on that given $n$ people at a party, it is possible to throw $1$ member out the window, leaving $n - 1$ members remaining.

## Problem 1

Prove that given $n$ attendees of a party, one can defenestrate $n$ members.

### Solution

Proof $\textbf{Lemma:} \textit{ Given integer }k; 0\le k\le n, k \textit{ members of a party of size }n.$

We use mathematical induction. We start with $k = 0$, which is trivial. We induct; give $i < n$ members defenestrated, $n - i$ remain. We throw out $1$ member, leaving us with $i + 1$ members defenestrated and $n - (i + 1)$ members remaining.

Thus lemma is true. Plugging in $k = 0,1,2,3\cdots$, we have $\mathbb{Q.E.D.}$

Try your hand at the following problems:

## Problem 2

Prove that given $n$ couples, you can defenestrate $1$ couple $n$ times.

## Problem 3

Jim and his wife Jeri attend a party with 4 other married couples. As they enter, Jim and Jeri shake hands with some of the guests, but not with each other. During the evening, each person except 1 shakes hands with some of the guests, but not with their spouse. After the party, Jim asks each guest how many people they shook hands with and got answers 0,1,2,3,4,5,6,7,8. How many people did Jeri shake hands with?