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You can do Monsky Monsky's theorem, that a square can not cannot be divided into an odd number of equal area triangles. On the way you will have to do

1. p-adic numbers,
2. Sperner's lemma,
3. present $\mathbb{R}$ as a vector space over $\mathbb{Q}$.

In case if you have more time, you could

• use Sperner's lemma to prove Brauwer's Brouwer's fixed point theorem.
• use (3) to do Dehn Invariants
• and I am sure you can find what to do with p-adic numbers
2 added 42 characters in body; added 62 characters in body

You can do Monsky theorem, that square can not be divided into odd number of equal area triangles. On the way you will have to do

1. p-adic numbers,
2. Sperner's lemma,
3. present $\mathbb{R}$ as a vector space over $\mathbb{Q}$.

In case if you have more time, you could

• use Sperner's lemma to prove Brauwer's fixed point theorem.
• use (3) to do Dehn Invariants
• and I am sure you can find what to do with p-adic numbers
1 [made Community Wiki]

You can do Monsky theorem, that square can not be divided into odd number of equal area triangles. On the way you will have to do

• p-adic numbers,
• Sperner's lemma,
• present $\mathbb{R}$ as a vector space over $\mathbb{Q}$.

In case if you have more time, you could use Sperner's lemma to prove Brauwer's fixed point theorem.
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