Navy Admiral's Logic Puzzles: Are You Smarter Than a Naval Commander?
Earlier today, readers were presented with three intriguing brainteasers designed to test logical reasoning and mathematical prowess. These puzzles, sourced from the new compendium Mathematical Puzzles and Curiosities by Ivo David, Tanya Khovanova, and Yogev Shpilman, challenge conventional thinking. Below, we revisit the puzzles with detailed solutions to see if you can outsmart a Navy admiral.
1. Battleships: The Single Ship Strategy
As an admiral overseeing a critical mission, you face a strategic choice. Option A involves deploying a single ship with a success probability of P percent. Option B entails sending two ships, each with a success probability of P/2 percent, where at least one must succeed for mission completion. Intuition might favor two ships for doubled chances, but the solution reveals otherwise.
Solution: Option A—sending a single ship—is consistently superior. For instance, if P equals 100 percent, the single ship guarantees success, whereas two ships each at 50 percent success yield only a 75 percent chance (since both fail with a 25 percent probability). Mathematically, let p represent the probability of success (P/100). With two ships at p/2 each, the probability that at least one succeeds is 1 – (1–p/2)2 = p – (p2)/4, which is always less than p. This counterintuitive result holds for all values of P, emphasizing that quality over quantity can be more effective in naval operations.
2. The Two Oracles: Distinguishing Randie and Rando
You encounter two oracles, Randie and Rando, who respond to yes-or-no questions. Randie answers randomly every time, while Rando randomly decides to tell the truth or lie per question, then answers accordingly. The challenge is to identify a method to tell them apart.
Solution: Yes, it is possible to differentiate them. The key is to ask Rando a question that forces a consistent response, such as, "Are you answering this question truthfully?" Both a liar and a truthteller will answer "YES" to this query. By repeating this question until receiving a "NO," you can deduce the oracle's identity. If a "NO" appears, it must be Randie, as Rando would always say "YES." If no "NO" emerges after multiple attempts, it is likely Rando. This clever approach leverages logical paradoxes to solve the puzzle.
3. Bad Maths: The Flawed Subtraction Trick
Johnny attempted a shortcut for subtraction problems, like 5548 – 5489 = 59, by canceling out digits. He tested it on a new calculation of the form XXYZ – XYZW, where X, Y, Z, and W are distinct digits, and found it resulted in XW. The puzzle asks how many digits in this new calculation match those in the original (i.e., X = 5, Y = 4, Z = 8, or W = 9).
Solution: Only Z and W correspond to the original digits, being 8 and 9, respectively. Breaking down the equation: 1100X + 10Y + Z – 1000X – 100Y – 10Z – W = 10X + W simplifies to 90X – 90Y = 9Z + 2W. Analysis shows W must be divisible by 9, so it is 0 or 9. If W = 0, then Z = 0, contradicting distinct digits. Thus, W = 9. Then, 9Z + 18 must be divisible by 10, leading to Z = 8. The equation reduces to 90X – 90Y = 90, so X = Y + 1, which has multiple solutions. However, W = 9 and Z = 8 are fixed, revealing the trick's limited applicability.
These puzzles, adapted for clarity, showcase the beauty of mathematical and logical thinking. The compendium offers a treasure trove of such challenges, encouraging readers to sharpen their minds. For more brainteasers, explore the ongoing puzzle series, which has been featured since 2015, and consider submitting your own ideas to contribute to the intellectual fun.
