Free the fly

Geometry Level pending

A fly is trapped inside a hollow cube. It moves from $A$ to $B$ along the edges of the cube, taking the shortest possible route. It then comes back to $A$ , again along the edges, taking the longest route (without going over any point more than once). If the total distance travelled is $504$ cm, what is the area of a face of the cube (in $cm^2$ )?

5184 7056 6241 3969

1 solution

Marta Reece
Jun 11, 2017

It takes $2$ edges for the shortest distance (blue) and $6$ for the longest (green).

(There are several ways the longest distance can be accomplished. Only one of them is pictured on the left.)

The total trip comes to $8$ edges.

The length of an edge is therefore $\dfrac{504}{8}=63$ cm.

Area of a face is $63^2=\boxed{3969}$ cm $^2$

Nice! With enough trial and error, we can see that the shortest route will take $2$ edges and the longest route $6$ edges. Is there perhaps a more rigorous way to prove these?

Zach Abueg - 3 years, 12 months ago

I have actually gone through all the possible paths to make sure I was right, but it's tedious and difficult to communicate. It is not hugely difficult though, as symmetry helps and many of the paths are obvious dead ends.

Marta Reece - 3 years, 12 months ago

Free the fly

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