🚧 Setup for 1.5

chapter_1/1_5/investigate.md

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+# Investigate!
+
+Q: Suppose there are 15 people at a party. Most people know each other already,
+but there are still some people who decide to shake hands. Is it possible for
+everyone at the party to shake hands with exactly three other people?
+
+A:
+
+Let $n$ be the amount of people at the party. Since every person must shake
+hands with exactly 3 people, then we can represent the amount of handshakes that
+takes place at the party to be $3n$.
+
+To determine if we have enough people at the party where everyone gets to shake
+the hand of exactly three other people, we can utilize what is known as the
+[Pigeonhole Principle](https://en.wikipedia.org/wiki/Pigeonhole_principle) where
+instead of pigeons we consider the amount of handshakes $3n$ and then evaluate
+the remainder of how many containers, or "handshake contributions" every
+handshake creates, in this case $2$.
+
+To understand this, consider that every time a single person shakes another
+person's hands, it counts as 2 handshakes:
+
+Alice shakes Bob's hand, but Bob also shakes Alice's hands. For the context of
+this problem, 2 handshakes occur every time two individuals shake hands.
+
+If we tally up the total amount of handshakes that must take place at this
+party:
+
+$$ 3n =  3 * 15 = 45 \text{ handshake contributions} $$
+
+And we look at the remainder of how many handshakes are contributed:
+
+$$ 45 \not\equiv 0 \mod 2$$
+
+We can see that we'll always need an even amount of people at this party for
+everyone to be able to shake the hands of exactly 3 people.