No Nuclear Disaster But...

[plainglazed]

When I turned on the lights in our triangularly shaped dog den I noticed what I think is a previously undiscovered survival

technigue of the common cockroach. Three such vermin had been feasting on a piece of kibble but as soon as the light came on

they all scattered, keeping both maximum distance from one another and taking the shortest straight line path to the edges of

the pen where they quickly discovered some crack and made their escape. Quite fantastic, really. But then I wondered, if the

pen has uniform side length of 2 meters, what is the expected average total distance the three pests travel given that piece

of kibble could be anywhere within the den?

[Guest]

Though an interesting problem, you need to give some clarity.

You have stated the roaches kept maximum distance between each other, this would be an angle between their paths of 120 degrees; however, you also stated shortest distance to the side, which would be perpendicular to the side, requiring only 90 degrees between the paths of the roaches. Obviously both are impossible!

[plainglazed]

Though an interesting problem, you need to give some clarity. You have stated the roaches kept maximum distance between each other, this would be an angle between their paths of 120 degrees, (correct) you also stated shortest distance to the side, which would be perpendicular to the side (correct), requiring only 90 degrees between the paths of the roaches (not correct).

[Guest]

It's been a while since I got the scientific calculator out but is it something like

3x (sin 60 + 1) = total distance travelled by all three roaches or average distance per roach is (sin 60 + 1)?

I think I may have misinterpreted your question somewhat but for clarity: I have assumed that they run to one of the sides hitting it at 90 degrees, then into one of the corners to escape?

If they only need to get to the side then remove the plus 1.

It is certainly true for the very centre and I'm guessing that it doesn't make any difference where in the den you start.

Edited June 29, 2011 by RESHAW

[Guest]

1.7320508 - missed the triangular part first time.

[k-man]

The combined distance travelled by all three roaches is always sqrt(3), regardless of where the kibble was.

[Guest]

it is the Sqrt of 3 which is 1.73205.....

make a triangle with lines going out from the center to each of the midpoints of the sides

Edited June 29, 2011 by harpuzzler

[plainglazed]

Kudos to thoughtfulfellow, k-man, and harpuzzler. k-man gets top marks for stating the distance is a constant as is easily proved by comparing the area of the entire equilateral triangle to the three triangles formed by connecting the random interior point to each vertex of the original triangle.

For the general case of an equilateral triangle ABC of side length s, the area = sqrt(3)s²/4 and the sum of the areas of triangles ACK, CBK, BAK = s/2(h₁+h₂+h₃). Equating the two yields h₁+h₂+h₃=sqrt(3)s/2.

Edited June 30, 2011 by plainglazed