Nichomauss' Theorem

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Nichomauss' Theorem

Nichomauss' Theorem states that $n^3$ can be written as the sum of $n$ consecutive integers, thus giving us $1^3+2^3+...+n^3=(1+2+...+n)^2$.

A Visual Proof

Imagine a cuboid with a height of $1$, and length and width of $n$. Divide it into unit cubes. Now, starting from the bottom right of the cuboid (when it's flat on the ground), imagine the first cube to have $l=w=h=1$ (call this $a_1$). Now, extend its length and width by $2$ to get another cuboid with $l=w=3$ and $h=1$ (call this $a_2$). Color $a_1$ yellow and $a_2$ blue. Now, it's easy to see that $a_2$ has $(3 \cdot 3 \cdot 1)-(1 \cdot 1 \cdot 1)=8$ unit cubes and $a_1$ has $1 \cdot 1 \cdot 1=1$ unit cube. We can then rearrange the $8$ unit cubes into a $2 \cdot 2 \cdot 2$ cube. Thus, we clearly have $(1+2)^2=1^3+2^3$. And we can continue this process to $n$ unit cubes and $n$ cuboids with $l=w=n$, $h=1$, which gets us to $(1+2+...+n)^2=1^3+2^3+...+n^3$.

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