This morning, three challenging puzzles were presented. Below are the problems restated along with their explanations.
1. Naval Strategy
As a fleet commander overseeing a critical operation, you face two options:
a) Deploy one vessel with a success probability of P percent.
b) Deploy two vessels, each with a success probability of P/2 percent. The mission succeeds if at least one vessel does.
Which choice offers better odds?
Explanation: Opt for the single vessel. While it may seem advantageous to use two, consider P equals 100: the single ship ensures success, but the pair only yields 75 percent likelihood (as both failing is 50 percent times 50 percent, or 25 percent). For any P, the single ship’s success rate exceeds that of the duo. Let p represent the success probability (P/100). For two ships at p/2 each, the probability both fail is (1 – p/2)^2, so at least one succeeding is 1 – (1 – p/2)^2 = p – (p^2)/4, which is less than p.
2. Distinguishing Oracles
You encounter two figures, Randie and Rando, who respond to queries with yes or no. Randie replies randomly each time. Rando randomly chooses to be truthful or deceptive per question and answers based on that.
Can you identify which is which? If yes, how?
Explanation: It is possible. Note that certain questions prompt Rando to always say ‘yes,’ like: ‘Are you being honest in this response?’ Both truthful and deceptive versions yield ‘yes.’ Pose this repeatedly until hearing ‘no,’ indicating Randie. Consistent ‘yes’ suggests Rando.
3. Flawed Calculation
A student computes 5548 minus 5489 as 59, reasoning that 548 cancels, leaving 59. Testing further, he subtracts XXYZ minus XYZW (where X, Y, Z, W are unique digits) and gets XW. How many digits match the original (is X=5, Y=4, Z=8, or W=9)?
Explanation: Z and W match, being 8 and 9. The equation simplifies to 1100X + 10Y + Z – 1000X – 100Y – 10Z – W = 10X + W, reducing to 90X – 90Y – 9Z – 2W = -10X – 2W + 10X + W, wait, actually from algebra: 90X – 90Y = 9Z + 2W, wait no, properly: it leads to W divisible by 9, so 0 or 9. W=0 implies Z=0, violating distinctness. Thus W=9, and Z=8 satisfies divisibility by 10 for 9Z + 18. Then 90X – 90Y = 90, so X = Y + 1, with multiple possibilities. Only Z=8 and W=9 align with the initial numbers.
These challenges come from the recent collection ‘Mathematical Puzzles and Curiosities’ by Ivo David, Tanya Khovanova, and Yogev Shpilman. The phrasing of the puzzles and answers has been adjusted for this feature. Puzzles appear here every other Monday since 2015. Suggestions for future ones are welcome via email.
