An ant is at a crossroads of a square grid of highways spaced 1 meter apart.

The ant can walk 1 meter in 15 seconds along a highway, but off-road it can only travel half as fast.

Find the area of the region composed of all points the ant can reach in at most 30 seconds.

Note: an ant highway is just a line with no thickness.

Consider the highway system:

+….+….a….+….+

+….b….+….h….+

c….+….A….+….g

+….d….+….f….+

+….+….e….+….+

Key: “A”=Ant; “+”= other crossroads; a, b,… h =

reachable highway-bound crossroad destinations.

Note: if Ant stayed strictly off-road diagonally, she could not reach “b”, “d”, "f" or "h", since the diagonal is of length 1.414m, far greater than 1m

reached off-road in 30 sec..

So the question becomes: if we consider square aceg as the possible outer area, where within can Ant

reach?

I am thinking now of Ant’s range being the superposition of circles formed by

sending Ant up a highway and then going off-road and any angle. The resulting loci of destinations will be something like

a 90 deg-tipped square with each side inwardly-bowed, like a superellipse.

TBC…

*Edited on ***August 17, 2018, 5:55 pm**