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Puzzles & Logic
Reasoning & Riddles
Classic puzzles and the logic that makes them work.
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A standard sudoku grid is what size?
Classic Puzzles and How They Work
Most durable puzzles are a piece of mathematics in disguise. Several of them founded entire fields once someone took them seriously.
About this topic
A puzzle earns its longevity by having structure underneath. The Tower of Hanoi is a recursion made physical; each extra disc doubles the work, which is why the legend about monks finishing the world-ending sixty-four-disc version is safe. The seven bridges of Konigsberg turned into the first theorem of graph theory when Euler ignored distances and shapes and looked only at what connects to what. Latin squares were a curiosity for a century and are now used to design experiments. That pattern repeats. The pigeonhole principle sounds too obvious to be worth stating and proves results that are not obvious at all. Nonograms and cryptarithms both reduce to constraint satisfaction, the same problem shape that scheduling software solves. These ten questions are about the mechanics rather than the trivia: what each puzzle actually asks and why the answer takes the form it does.
Questions & answers
Classic Puzzles and How They Work: all 10 questions and answers
Every question in this quiz is listed below with its correct answer and the reasoning behind it. Play first if you would rather not see the answers — or read through as a study sheet.
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1In the Tower of Hanoi, which rule must never be broken?
- A No disc may return to the peg it started on
- B The smallest disc may not move twice in a row
- C Discs may only move to the right
- D A larger disc may never rest on a smaller one Correct
Answer: D. A larger disc may never rest on a smaller one
Why: The minimum number of moves is two to the power of the number of discs, less one. Each additional disc therefore doubles the entire task.
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2A magic square is arranged so that what is true?
- A All four corners are prime numbers
- B Each row reads the same in both directions
- C Every row, column and main diagonal adds to the same total Correct
- D Every number appears exactly twice
Answer: C. Every row, column and main diagonal adds to the same total
Why: The oldest known example is the Chinese Lo Shu square. It is three by three and every line totals fifteen.
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3In the Monty Hall problem, switching doors after a losing door is opened gives what chance of winning?
- A Three in four
- B Two in three Correct
- C One in two
- D One in three
Answer: B. Two in three
Why: The host never opens the prize door, so his choice carries information. The puzzle collapses to an even split only if the door he opens is chosen at random.
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4The pigeonhole principle states what?
- A If there are more items than containers, at least one container holds more than one item Correct
- B Every list can be sorted in a single pass
- C Any two people share a birthday
- D Random outcomes even out given enough trials
Answer: A. If there are more items than containers, at least one container holds more than one item
Why: It sounds too obvious to be useful and is not. It is enough on its own to prove that two people in London have exactly the same number of hairs on their heads.
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5The Seven Bridges of Konigsberg problem asked whether a walk could do what?
- A Visit each island twice
- B Return home by the shortest possible route
- C Cross the river without using a bridge
- D Cross every bridge exactly once Correct
Answer: D. Cross every bridge exactly once
Why: Euler proved it impossible in 1736 by reducing the city to points and connections. That step is generally taken as the beginning of graph theory.
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6In a cryptarithm such as SEND + MORE = MONEY, what do the letters stand for?
- A Chemical elements
- B Coordinates on a grid
- C Digits, with each letter representing a different one Correct
- D Words hidden inside a message
Answer: C. Digits, with each letter representing a different one
Why: The usual additional rule is that no number may begin with zero. That constraint is often the wedge that opens the whole puzzle.
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7In a classic river-crossing puzzle, what is the usual constraint?
- A Every item must cross the river twice
- B Certain items cannot be left together on a bank unattended Correct
- C The boat sinks after three crossings
- D The river may only be crossed after dark
Answer: B. Certain items cannot be left together on a bank unattended
Why: The wolf, goat and cabbage version appears in a manuscript attributed to Alcuin of York around the year 800. Solving it requires taking something back, which is the step most people resist.
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8A Latin square is a grid in which what holds?
- A Each symbol appears exactly once in every row and every column Correct
- B Every row adds to the same total
- C The grid looks identical after any rotation
- D Symbols alternate between two types
Answer: A. Each symbol appears exactly once in every row and every column
Why: Euler studied them in the eighteenth century and gave them the name. They are now used to design experiments in which every treatment must meet every condition once.
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9In a nonogram, what do the numbers beside each row tell you?
- A The sum of the digits in that row
- B How many cells must be left empty
- C Which colour each cell takes
- D The lengths of the runs of filled cells, in order Correct
Answer: D. The lengths of the runs of filled cells, in order
Why: Solvers progress by overlapping the leftmost and rightmost legal placements of a run. Any cell filled in both extremes must be filled in the solution.
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10A lateral thinking puzzle is designed to be solved by doing what?
- A Trying every option until one works
- B Recalling a specific piece of general knowledge
- C Questioning an assumption the puzzle quietly invites you to make Correct
- D Applying a standard formula
Answer: C. Questioning an assumption the puzzle quietly invites you to make
Why: Edward de Bono popularised the term in the 1960s. The best examples turn on exactly one buried assumption rather than a chain of steps.
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