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Take an equilateral triangle with side length $2^n$. Divide it up into equilateral triangles with side length 1, and delete the top triangle. Call this shape $T_n$:

Take three of the smaller equaliteral triangles, and join them together to form the following shape:

Formed shape

Prove that you can tile $T_n$ with tiles of this shape for every $n\in\Bbb N$.

I prove for the base case but I don't know how to continue with inductive steps. Want some way of 'relating' $T_{n+1}$ and $T_n$; that is, some way of turning the task "cover an equilateral triangle with side length $2^{n+1}$ with our tiles" into the task "cover an equilateral triangle with side length $2^n$ with our tiles."

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Here is the answer to the intended question:

$T_1$ is just one of our small three-triangle tiles; $T_n$ can be formed from four copies of $T_{n-1}$ and one more small tile as shown above, so can be tiled entirely with small tiles. The arrangement of the small tiles is reminiscent of the chair substitution.

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  • $\begingroup$ Thank you for the editing! I'm so sorry there was a huge typo which impacts the correct answer. The equilateral triangle with side length "2n" should be 2^n. And I am still confused on how to prove Tn⇒Tn+1. $\endgroup$ – Duncan Alf Zhang Sep 16 '17 at 12:36

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