At the time of this writing, the United States has launched and landed four types of manned spacecraft. There have been three types of capsules (Mercury, Gemini and Apollo) and one orbiter-glider (Space Shuttle).
Based on the statistical analysis of the American spacecraft experience, which one is safer in terms of risk of a fatal accident — capsules or orbiter-gliders? How confident are you of your answer? If you are a truly wild Bayesian at heart, you may try to include experience from the Russian capsules (Vostok, Voskhod and Soyuz) and the Chinese capsule (Shenzhou).
Know the answer? Send your solution to email@example.com.
Following is a very thorough solution submitted by Bob Conger.
Based on the +/-2 inexactness of the lock mechanics stated in the problem, it will be sufficient to try combinations consisting only of 0, 5, 10, 15, 20, 25, 30 and 35, since all other values from 0 to 39 are within +/-2 of one of these eight values.
Assuming that two or even three of the values in the combination could be (or “round to”) identical to one another, the potential number of combinations to try is 83, or 512. The mechanics of the lock possibly do not permit repeating the same exact or approximate number in a row, in which case the number of combinations to try is 8x7x7, or 392.
However, we can effectively test eight (or seven) combinations in one fell swoop dialing the first two numbers, say 25-30, in the usual way and then turning clockwise to 25, testing (gently) to see if the lock opens; continuing clockwise to 20, testing to see if the lock opens; continuing clockwise to 15, testing to see if the lock opens; and so on until reaching 30 and testing to see if the lock opens. By following this procedure for all possible first-digit, second-digit possibilities, the time required is only slightly greater than trying 64 (or 56) full combinations.
To gain a bit more time efficiency (by trying the quicker combinations first), I would start by trying all the combinations where the second number of the combination is equal to the first number of the combination + 5, mod 40. So, if the first number is 5, the second number is 10. If the first number is 35, the second number is 0. These combinations require turning the dial only 9/8 of a spin to get from the first number to the next. Then, I would gradually work the dial clockwise, testing at every multiple of five whether the lock would open, starting with 5-10; or 10-15; or 15-20; or 20-25; or 25-30; or 30-35; or 35-0; or 0-5.
Then, I would try all the combinations where the second number of the combination is equal to the first number + 10, mod 40, and so forth, finishing up with the combinations where the second number is equal to the first number + 40, mod 40, which is equal to the first number of the combination. Or, if the mechanical design of the lock is such that the first and second numbers cannot be (approximately) equal, I would finish up with the combinations where the second number is equal to the first number + 35, mod 40.
This problem reminded me of my days in high school, when some of my friends and I discovered that the mechanical design and behavior of Master padlocks allowed for a dramatically reduced number of possible combinations to open them. I don’t recall that we reduced the number quite as effectively as the following tutorials,* but it started with a similar process of deducing one of the numbers of the combination by simple mechanical means.
I saw another post whose author claimed to be able to reduce the number of combinations down to eight possibilities by identifying numbers where the dial catches or has resistance.
Solutions were also submitted by Brian Barsotti, Jordan Bonner, Kristen Fox-Neff, Clive Keatinge, Richard Kollmar, John Pagliarulo, Hannah Park, Brad Rosin, Eric Savage and Betty-Jo Walke. Jerry Miccolis was inadvertently left off of last issue’s list of solvers for “Risk Appetites”, the 2018 September/October AR puzzle.