*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?

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

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

### Know the answer? Send your solution to ar@casact.org.

## 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:

Program |
Manned Flights |
Fatal Incident Flights |
Fatal Flight Rate |

Mercury |
6 | 0 | 0% |

Gemini |
10 | 0 | 0% |

Apollo (including Skylab crews and Apollo-Soyuz) |
15 | 0 | 0% |

Total |
31 | 0 | 0% |

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:

Program |
Manned Flights |
Fatal Incident Flights |
Fatal Flight Rate |

Vostok |
6 | 0 | 0% |

Voskhod |
2 | 0 | 0% |

Soyuz |
141 | 2 | 1.4184% |

Shenzhou |
6 | 0 | 0% |

Non-U.S. Capsule Total |
155 | 2 | 1.2903% |

Including U.S. Capsule Total |
186 | 2 | 1.0753% |

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.