Navy Admiral's Puzzle: Are You Smarter Than a Naval Commander?
Earlier today, readers were challenged with three intriguing puzzles designed to test logical reasoning and mathematical skills. These brainteasers, sourced from the new compendium Mathematical Puzzles and Curiosities by Ivo David, Tanya Khovanova, and Yogev Shpilman, have been adapted for clarity. Below, we present the solutions with detailed explanations.
1. Battleships: A Probability Conundrum
As an admiral in the Navy, you face a critical mission decision. You have two options:
- Option A: Send a single ship with a success probability of P per cent.
- Option B: Send two ships, each with a success probability of P/2 per cent, where at least one must succeed for mission success.
Intuition might suggest that two ships offer better odds, but the solution reveals otherwise. For instance, if P equals 100, Option A guarantees success, while Option B yields only a 75 per cent chance, as both ships fail with a 25 per cent probability (50 per cent x 50 per cent).
Mathematically, let p represent the probability of success (P/100). For two ships with probability p/2 each, the chance that both fail is (1–p/2)2. Thus, the probability of at least one success is 1 – (1–p/2)2 = p – (p2)/4. This expression is always less than p, confirming that sending a single ship is the superior strategy for all values of P.
2. The Two Oracles: A Logic Challenge
You encounter two oracles, Randie and Rando, who answer yes or no to any question. Randie responds randomly to all queries, while Rando randomly decides to tell the truth or lie for each question and answers accordingly.
Is it possible to distinguish between them? Yes, by exploiting a key insight. You can ask Rando questions that guarantee a "YES" response, such as "Are you answering this question truthfully?" Both a liar and a truth-teller would answer "YES" to this query.
By repeatedly asking this question until you receive a "No," you can identify Randie, as Randie might answer "No" randomly. If you never get a "No," it is highly likely you are dealing with Rando.
3. Bad Maths: A Subtraction Trick
Johnny's homework involved calculating 5548 – 5489, which equals 59. He noticed that the digits 548 seemed to cancel out, leaving 59. To test this, he performed a subtraction of the form XXYZ – XYZW, where X, Y, Z, and W are distinct digits, and found it resulted in XW.
The question is: How many digits in this new calculation match those in the original (i.e., does X = 5, Y = 4, Z = 8, or W = 9)? Breaking it down, the expression is 1100X + 10Y + Z – 1000X – 100Y – 10Z – W = 10X + W, simplifying to 90X – 90Y = 9Z + 2W.
From this, we deduce that W must be divisible by 9, so it is either 0 or 9. Also, 9Z + 2W must be divisible by 10. If W = 0, then Z = 0, contradicting the distinct digit requirement. Therefore, W = 9. Substituting, 9Z + 18 must be divisible by 10, leading to Z = 8. This leaves 90X – 90Y = 90, so X = Y + 1, which has multiple solutions. Thus, Z and W are 8 and 9, matching the original digits.
These puzzles are part of a broader collection in Mathematical Puzzles and Curiosities, edited for this column. The author has been setting puzzles on alternate Mondays since 2015 and welcomes suggestions via email.
