Don't Think Outside The Box!

Geometry Level 2

Brilli the ant is trapped on a cube-shaped planet. She wants to go from one corner [point $A$ ] to the opposite corner [point $B$ ]. However she wants do this in a way such that she has to cross the shortest distance possible. If the length of the sides of the cube is $1$ , the shortest distance between $A$ and $B$ can be expressed as $p+\sqrt{q}$ where $p$ and $q$ are non-negative integers and $q$ is square-free. What is $q-p$ ?

Details and assumptions :

Brilli the ant is completely confined to the surface of the cube. She can't move inside or outside the cube.

The answer is 5.

2 solutions

Mursalin Habib
Apr 27, 2014

Try opening up the cube like this:

Imgur Imgur

It is apparent that the shortest distance between $A$ and $B$ is $\sqrt{2^2+1^2}=\sqrt{5}$ .

That means $p=0$ and $q=5$ and $q-p=5-0=\boxed{5}$ .

Ugh, I was pondering for 5 minutes why $\sqrt{5}$ wasn't the answer, then I realized it was, because $p=0$ . Trippy.

Daniel Liu - 7 years, 1 month ago

A common wrong answer is $1+\sqrt{2}$ . That's why I put $p+\sqrt{q}$ as the format.

Mursalin Habib - 7 years, 1 month ago

I did so,and broke my head!

Ralph Mackenson Lefruit - 5 years, 1 month ago

I was also hesitant while entering $\sqrt{5}$ . And I had to recheck all over again, cause of the format of the answer

Vishal Sharma - 7 years, 1 month ago

Exactly! It's so throw-y off-y! :O

Finn Hulse - 7 years, 1 month ago

This is a famous problem created by Henry Ernest Dudeney From The Canterbury Puzzles 1908. The original problem is called the Spider and the Fly Problem. http://mathworld.wolfram.com/SpiderandFlyProblem.html

Satyen Nabar - 7 years, 1 month ago

Thanks for sharing

Krishna Ar - 7 years ago

really out of box.....

Max B - 7 years, 1 month ago

i assumed she could move on the edge of the box so my answer came out to be 1 instead

Harsh Kumar - 7 years, 1 month ago

She can, but she wants to cover the shortest distance possible and that doesn't happen when she moves along the edges.

Mursalin Habib - 7 years, 1 month ago

Sorry , its"down the edge to B".

Keshav Kumar - 7 years, 1 month ago

SAME THING

Pi Han Goh - 7 years, 1 month ago

Wait a minute you said ant is trapped on planet so how are you unfolding it.ant will have to travel on diagonal line of upper surface and then down the edge to point A.so answer is 1+2^(1/2).now 2-1=1.

Keshav Kumar - 7 years, 1 month ago

She is trapped! She is confined to the surface of the cube.

It's easier to visualize the whole thing when you unfold it. But Brilli's path is still along the surface of the cube for this solution.

Mursalin Habib - 7 years, 1 month ago

See the shortest distance should be sqrt(2)+1

Aishwary Omkar - 6 years, 7 months ago

Finn Hulse
May 1, 2014

Here is a very similar problem by me, Cupid the Caterpillar . As @Mursalin Habib mentioned, drawing a net is the easiest possible way.

Don't Think Outside The Box!

The answer is 5.

2 solutions

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