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\tutorial{tutorial2}{Problems for Tutorial 2}{February 18, 1997}

\section{Warmup}

Prove by induction on $k$ that 
$$
5 \mid  (3^{2k} + 4^{k+1})
$$
for any integer $k\geq 0$. 

\section{Peano Strikes Back}

Recall that addition was defined by:

D1)  $ \forall n \in \naturals, n+0 = 0+n$ 

D2)  $ \forall n,m \in \naturals, n+m' = (n+m)'$

Prove by induction the cancelation law of addition. 

$$
 \forall n,m,k \in \naturals, n+k = m+k \implies n = m
$$ 


\section{Handshaking Problem}

I recently went to a party with my spouse; there were 11 other
married couples there.  A lot of handshaking happened (but of course
no one shook hands with their spouse).  At the end, it turned out that
for every $k$ from $0$ to $22$, someone other than me had shaken hands
with exactly $k$ people.  How many people did I shake hands with?
Prove it.

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