# This ant sure is smart. But how fast is it?

A mad scientist uses ethically questionable methods to study the spatial cognition of ants.

Last time, he left his lab ant in a cubic room and filled it with painfully intoxicating gas, capable of killing an adult ant in exactly twenty minutes.

The room was perfectly sealed except for a small hole in the (diametrically) opposite vertex from that on which the ant was sitting.

However, using it's notorious spatial intuition, the ant was able in no time to compute the shortest path and run for it's life. Unfortunately the ant could not survive for much longer after escaping, due to permanent spiracle and kidney toxicity.

Even then, the scientist's controversial experiment had repercussions all over the world, because it proved that the solution previously computed by mathematicians was indeed optimal, and no path could be shorter than $\sqrt5$ times the side length of the box.

The mad scientist then made a new room to defy any kind of geometrical intuition from his ants.

This time, he created a 98-dimensional hypercube which could only be walked along the inside of it's 97-dimensional hypersurface. Then he filled it's entire $1\;googol$$\:\mathrm{dm^2.m^{96}} hypervolume with a toxic hypergas capable of killing an ant in the same amount of time as before. However, the next lab ant noticed that with simple spatial reasoning and dimensional analogy it's again possible to derive the shortest path, and started in no time to follow it, at maximal speed, to the opposite vertex of the hypercube. Very unluckily for the ant, it died exactly as it came out of the hyper-room. The ant was very smart for figuring out the optimal solution, but the scientist was soon asking himself if the ant died because it was too slow, or the time was too little. According to this experiment, how fast was the ant? (Answer in centimeters per second, with at least two decimal digits) • It's a bit unclear what you're asking. What units would the answer have? – MisterEman22 Oct 2 '15 at 18:43 • I just forgot the units. Had centimeters per second in my mind, but in no way you could deduce it from what was given in the question. My bad – MathET Oct 2 '15 at 18:48 • I get why the dimension was chosen to be 98, but why was the room's hypervolume chosen to be a googol? If the hypervolume was instead\frac{\text{googol}}{100}=10^{98}, then the answer would be exactly 5 meters per minute. – Mike Earnest Oct 2 '15 at 21:59 • Mostly because it sounded good :). But I see how an exact answer would be better. I edited now, should I update your answer too? – MathET Oct 2 '15 at 22:38 • Your call. You are right that saying something has a volume of googol sounds way cooler B) I'm also impressed anytime I see a puzzle in more than 4 dimensions which is doable, thanks for the post! – Mike Earnest Oct 3 '15 at 0:25 ## 1 Answer First, here is the reason the answer is \sqrt{5} in the three dimensional case. Take one of the rooms vertical walls containing the opposite corner, and fold it down, so it forms a 2 by 1 rectangle with the floor. The ant's optimal path is to walk the diagonal of this rectangle, of length \sqrt{2^2+1^2}. Given a d dimensional cube, imagine folding down one of the walls to form a (d-1) dimensional hyperrectangle with dimensions 2\times 1\times 1\times \dots\times 1. The ant's optimal path is to walk the diagonal of this rectangle, which is of length \sqrt{2^2+1^2+\dots+1^2}=\sqrt{d+2}. The side length s of the room is given by s^{98}=10^{100}dm^2m^{96}$$=10^{98}m^{98}$, meaning $s=10$. So that in this case the optimal path length is $$\ell=\sqrt{98+2}\cdot 10=100\text{ m}$$ Since the ant traverses this length in exactly 20 minutes, his speed must be $$\frac{100}{20}\text{ m}/\text{min} \approx 8.33\text{ cm}/\text{s}$$

• How did you get the 20 minutes? – JonTheMon Oct 2 '15 at 21:06
• @JonTheMon "capable of killing an adult ant in exactly twenty minutes", "capable of killing an ant in the same amount of time as before.", "Very unluckily for the ant, it died exactly as it came out of the hyper-room." – warspyking Oct 2 '15 at 21:39
• @warspyking Bah, glossed over it b/c it was spelled out. – JonTheMon Oct 2 '15 at 21:44