Lawn Mower Geometry

This puzzle was created and written by Don Mango. Neither Don nor I have solved it. So, it is the duty of you, the dear readers of this esteemed column, to solve it by any means necessary!

When I was growing up in Texas, I would cut lawns in the summer — a physically draining experience. As I slogged along, I was naturally thinking of how to optimize my cutting by minimizing the time to completion. I had all sorts of half-baked heuristics about how to handle trees or long skinny patches or even triangles. Over the years, I have noodled about the possibility of some very smart actuaries being able to systematically approach the problem. So, I humbly offer the problem of Lawn Mower Geometry.

Let’s assume the following:

• A nice riding mower with a 3-foot diameter (perfect circle) travels at 10 feet per second.
• It can turn on its center (rotate) at 45 degrees per second.
• Stopping costs four seconds (the combination of slowing down, turning and accelerating again).
• There are no issues with the boundaries of the lawn and no weed whacking is necessary. All the grass has to be cut with the mower.

Problem: What is the fastest way to cut each of these shapes?

1. A 50-foot-by-100-foot rectangle.
2. A 50-foot-(side A)-by-100-foot-(side B)-right triangle.
3. Two different trapezoids:

• Attached on the 50-foot side.
• Attached on the 100-foot side.

Manned Spaceflight Safety

Note to readers from Jon Evans: As of July 17, 2019, we received no solutions for the “Manned Spaceflight Safety” puzzle. This was actually a very easy puzzle, but it required spending just a small amount of time looking up some statistics using Google, Wikipedia or another online reference source. There seems to be a pattern that whenever a puzzle requires some amount of thought or online information lookup involving engineering, natural sciences or other real-world subjects, readers tend not to submit solutions. Our advice is not to panic or despair when you see such a problem, but calmly and methodically think through the problem and about how readily available information might be used to solve it.

The puzzle asks solvers to use statistical analysis of the American spacecraft experience to determine whether capsules or orbiter-gliders are safer in terms of risk of a fatal accident. Data for this problem, as of July 17, 2019, is available online at

www.wikipedia.org/wiki/List_of_human_spaceflights

and

www.wikipedia.org/wiki/List_of_spaceflight-related_accidents_and_incidents.

Data for the U.S.-manned capsule flights is as follows:

These tabulations do not include the two flights of the North American X-15 and the three flights of the private SpaceShipOne — both are rocket-powered airplanes that technically entered space-level altitudes, but at only a fraction of orbital velocity. Readers may also recall the tragic fatalities of the Apollo 1 test in 1967, but that was a ground test inside a capsule and not an actual flight test.

Now, for the Space Shuttle orbiter-glider, the data is as follows: 135 manned flights and two fatal incident flights for a fatal flight rate of 1.4815%.

We could construct many statistical tests, but it is hard to think of a good test that shows this experience implies any statistically meaningful difference between the capsules and the orbiter-glider. Perhaps the simplest approach is to assume that each flight for the capsules was an independent Bernoulli trial and that the probability of each capsule flight being a fatal incident was the same as the average rate for the shuttle. Under those assumptions, the probability that none of the 31 capsule flights would be a fatal incident flight, as none were, would be (100% – 1.4815% )³¹ ≈ 63%.

Now for the truly wild Bayesian at heart, let’s suppose we want to include the Soviet/Russian and Chinese capsule experience:

If we assume again that each capsule flight had the same 1.4815% fatal probability as for the Space Shuttle experience, then the probability (calculated using a binomial distribution) of two or fewer fatal flights among the 186 total international capsule flights is about 48%. So, again the historical experience suggests no meaningful difference in fatal flight rates between capsules and orbiter-gliders.