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The Cantor Set

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Liam from Wilbarston School sent us this well-explained solution:

Only the middle thirds of line segments are removed, so endpoints of line segments will not be removed.
Therefore, all I need to do is determine which are endpoints and which are not.

$\frac{1}{3}$ is an endpoint of C2, so it is in the Cantor set.

$\frac{3}{81}$ = $\frac{1}{27}$

$\frac{1}{27}$ is an endpoint of C4, and therefore $\frac{3}{81}$ is in the Cantor set.

$\frac{4}{9}$ is between $\frac{1}{3}$ and $\frac{2}{3}$, thus it is in the middle third of C1 and is removed from C2 and as such isn't in the Cantor set.

$\frac{4}{81}$ is between $\frac{1}{27}$ and $\frac{2}{27}$ and therefore is in the middle third of C3.
It is removed from C4 and therefore does not belong.