Ant got lost again!

Geometry Level 4

An ant is walking on a very long piece of wood with width 1. The ant wants to walk out of this wood by following a predetermined path, which guarantees that the ant will be able to eventually get out.

Let $L$ denote the minimum length of such a path.

Which of the following inequalities must be true?

\frac57 \pi \leq L< \pi

2\sqrt5 \leq L \leq 2 \sqrt6

\pi \leq L \leq 2 \sqrt3

1 solution

Ossama Ismail
Jan 5, 2017

Answre is $\large c$

There are three possible solutions for this problem (I am sure there may be much better solutions ):

1- The $Ant$ may walk in a circle with radius $\large \frac{1}{2}$ and get out in $\large \pi$ units at most.

2- The $Ant$ may move $2$ units at most from its location and get out if it walks distance $\large \sqrt 2$ units in any direction and then turns left or right and walk another distance $\large \sqrt 2$ . This make max distance $=\large 2 \sqrt 2$ .

3- The $Ant$ may move $\large \frac{2}{\sqrt3}$ units in any direction and then turns with an angle $= 120^o$ and walk another $\large\frac{2}{\sqrt3}$ . The make max distace will be $\large \frac{4}{\sqrt3}$ and the $Ant$ get out after walking distance $= \large \frac{4}{\sqrt 3}$ at most.

How do you know that we cannot do better than $\frac{ 5}{7} \pi$ ?

Calvin Lin Staff - 4 years, 5 months ago

Because the best results I obtained was 2.28. I don't have a concrete prove right now. I think with some geometry and optimization we may get a better answer. I will work on it when I have time.

Ossama Ismail - 4 years, 5 months ago

Can you prove that one of these three must be optimal?

Geoff Pilling - 4 years, 5 months ago

I am working on finding an optimal solution. I already have a solution better than 3 !!

Ossama Ismail - 4 years, 5 months ago

It seems to me that your option (3) would be hard to beat, but can it be proven?

Geoff Pilling - 4 years, 5 months ago

Ant got lost again!

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