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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
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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
show/hide this revision's text 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.