Solving a Truth-and-Lie Puzzle from Relative Cat Counts
Summary
This recreational logic puzzle assigns different cat counts to three speakers. Each tells the truth when addressing someone with fewer cats and lies when addressing someone with more cats. The accepted solution first derives the ordering of their counts from the rules governing the first statement, then uses the truth conditions of the next two statements to express the counts as fixed ratios and an average.
The final statement supplies an inequality: one count can exceed another by no more than three. Combining it with the ratios gives an upper bound for Bronya’s count, and requiring whole-number counts identifies a single solution in the answer. The response notes an ambiguity if two people have equal counts, since the puzzle’s speaking rule does not specify how ties work. This is a self-contained deduction exercise rather than a trading or quantitative finance method.
Key ideas
- The truth rules determine which statements must be true or false from the speakers’ relative counts.
- The first statements establish an ordering and ratios among the three counts.
- The last statement bounds the difference between two counts.
- Combining the ratios, bound, and integer requirement yields the stated solution.
- The puzzle leaves ties ambiguous under its rules.
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Full text
# Number of cats - brainteaser # Number of cats - brainteaser Asta, Bronya, and Clara all have different numbers of cats. They had the following conversation: i. Bronya to Clara: You have the most cats. ii. Asta to Bronya: I have exactly 30% more cats than you. iii. Asta to Clara: The number of cats you have is the average between the number of cats I have and the number of cats Bronya has. iv. Clara to Asta: You have at least 4 more cats than me. However, not all of these statements are true. When speaking to someone with fewer cats, the speaker always tells the truth. Otherwise when speaking to someone with more cats, the speaker always lies. How many cats does Bronya have? Is it 5 cats? Can someone tell me the answer? I have a hard time figuring this out. ## Answer by Attack68 (score 6) https://quant.stackexchange.com/a/80808 It is a stated axiom that when one party speaks to another who is observed to have a higher quantity the statement made is false. i) B speaks to C: "You have the most cats". This cannot be true, since if it were it would not satisfy the axiom. And if #C < #B then B would not lie. Hence one concludes: #B < #C < #A. ii) A to B: "I have exactly 30% more cats than B". This must be true from above: #B < #C < #A = 1.3#B. iii) A to C: "You have the average of A and B". This is also true as above: #B < #C = 1.15#B < #A = 1.3#B iv) C to A: "You have at least 4 more cats than me". This is untrue implying that #A <= #C + 3. Thus 1.3#B <= 1.15#B + 3, which implies 0.15#B <= 3 or #B <= 20. Working with integers means: (B,C,A) = (20, 23, 26). There is no other integer combination that satisfies all information. [Note if #C = #A, then it is an ambiguous definition under the terms of the question if "the most cats" still applies to either party, and also when A speaks to C she will lie or tell the truth] ## Answer by KaiSqDist (score 2) https://quant.stackexchange.com/a/80807 From statements (i) and (ii), the order of cats is - Asta > Bronya/Clara. From statement (iii), if Asta has 130% and Bronya is 100%, then Clara has 115%. From statement (iv), Asta has either 3, 2 or 1 more cat than Clara. With all these information, a sensible answer is Asta 26, Bronya 20 and Clara 23 cats that fulfills all the not-true and true statements above.
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