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Starvard University · Puzzles and Logic
Solve Dudeney's money, dissection, and chessboard puzzles with every answer shown, then learn Lewis Carroll's logic game of cakes, counters, and diagrams.
Hello, my name is Finn, and I'm going to help you understand this course.
Here it is, explained like you are 10:
A puzzle is a small problem with a secret door. At first it looks locked, then you find the one idea that opens it, and it feels like magic. In this course you will untie money puzzles, cut a triangle and rebuild it as a square, park queens on a chessboard so none can attack another, and learn the logic game invented by Lewis Carroll, the man who wrote about Alice. You will also learn why some arguments only look correct, and you will finish by inventing a puzzle of your own, with a solution you have tested yourself. Bring paper, scissors, and patience.
Henry Dudeney, an English clerk who became the most famous puzzle maker of his time, filled a book called Amusements in Mathematics with hundreds of these small locked doors, and this course walks through his best rooms: money, cutting out shapes, the chessboard, and moving counters across rivers. His rule was simple. The conditions of a puzzle should be easy to understand, even when the solution is hard, and a near answer is no answer at all. He also loved a good story around a puzzle, and his earlier Canterbury Puzzles are the book this one keeps pointing back to.
Then we change guides. Lewis Carroll, who taught mathematics at Oxford when he was not writing about Alice, turned logic itself into a board game with counters, cupboards, cakes, dragons, and Scotchmen. You will learn his diagrams well enough to test an argument the way you would test a bridge, by loading it and watching where it creaks. Every puzzle in this course comes with its full solution worked out in the lesson, because a puzzle that hides its answer teaches nothing. Bring a friend along if you can, because half of these puzzles are twice as much fun argued over a table.
The whole course in brief
What you will be able to do after this course
The seven lessons
Lesson 1
A puzzle is a question with the fat trimmed off. Everything you need is in front of you, nothing is hidden except the idea, and the answer can be checked the moment you find it. That is why puzzles train a mind the way scales train a musician. Henry Ernest Dudeney collected hundreds in Amusements in Mathematics in 1917, in families such as money puzzles, dissections, chessboard problems, and river crossings. He warned that they are deliberately not arranged in order of difficulty, so you never know whether one is as simple as it looks or has a pitfall in it.
Try one now, with its solution. Place the numbers 1 to 9 in a three by three grid so every row, column, and diagonal adds to the same total. One solution is 8, 1, 6 across the top, then 3, 5, 7 across the middle, then 4, 9, 2 along the bottom, and every line adds to 15. You can also prove the centre must be 5. The numbers add to 45, so each row adds to 15. The four lines through the centre add to 60, and they count every number once, plus the centre three extra times. So 45 plus three times the centre equals 60, and the centre is 5.
Lesson 2
Dudeney opens with money puzzles. His first, A Post-Office Perplexity, gives a customer a crown, five shillings or sixty old pennies, and asks for some twopenny stamps, six times as many penny stamps, and the rest in twopence-halfpenny stamps. Call the twopenny stamps x. The penny stamps are 6x, so both sorts together cost 8x pence, and the rest must complete 60. Try x equals 5: that is 5 twopenny and 30 penny stamps costing 40 pence, leaving 20 pence, exactly 8 stamps at twopence-halfpenny. No other value of x splits the remainder evenly.
His Cyclists' Feast is the same game. The bill is four pounds, or 80 shillings, two men slip away, and each man left pays two shillings more than his fair share. If 10 men started, the share was 8 shillings and the 8 who stayed paid 10 shillings each, exactly two more, so the answer is 10. The famous trap is his Two Aeroplanes. A man sells two machines for 600 pounds each, losing 20 percent on one and gaining 20 percent on the other. Most people guess he broke even. He did not: the first cost 600 divided by 0.8, or 750 pounds, the second 600 divided by 1.2, or 500 pounds. He spent 1,250, took back 1,200, and lost 50 pounds, because the percents acted on different costs.
Lesson 3
A dissection puzzle asks you to cut one shape into pieces that fit together to form another. Dudeney calls the class very ancient, known in China thousands of years ago, and insists on two rules: the solution must be geometrically exact, never near enough, and you may turn pieces over unless the puzzle forbids it. The key tool is Pythagoras. Build squares on the sides of a right angled triangle with sides 3, 4, and 5, and the small squares hold 9 and 16 cells while the large one holds 25, and 9 plus 16 equals 25. Areas add, so sizes can be settled before you cut.
His favourite family is the Greek cross, five equal squares joined crosswise. The old Hindu problem cuts it into five pieces to form a square. Dudeney does it in four pieces, even in four pieces all the same size and shape. His proof: draw the finished square on transparent paper, lay it over the cross at the right angle, and slide it about. Every position gives a different four piece solution, so there are infinitely many. His most famous feat is the Haberdasher's puzzle, published in 1902 as number 26 of his Canterbury Puzzles: cut an equilateral triangle into only four pieces that form a perfect square. Anyone could manage five. Four, its pieces hinged to swing between the two shapes, is the genius.
Lesson 4
You do not need to play chess for chessboard puzzles, only to know how the pieces move. Dudeney's Eight Queens, first proposed in 1850, asks for eight queens so no queen attacks another along a row, column, or diagonal. Here is one full solution, giving the column for each row from the top: 1, 5, 8, 6, 3, 7, 2, 4. Check any two queens: they never share a line. Dudeney records 12 fundamentally different solutions, making 92 counting turns and mirror images. His Eight Rooks is the opposite trick: rooks down the long diagonal attack every square, since every square shares a row with one, yet none guards another, diagonal neighbours sharing no row or column.
Counting is a puzzle too. How many rectangles does a chessboard contain? Choose any two of its 9 vertical lines and any two of its 9 horizontal lines and you have one rectangle: 36 times 36, or 1,296 rectangles, of which 204 are squares, 64 of one size, 49 of the next, and so on down to 1. In the river problem Dudeney traces to Alcuin, a man must ferry a wolf, a goat, and a cabbage across one at a time, never leaving wolf with goat or goat with cabbage. Solution: take the goat, return alone, take the wolf, bring the goat back, take the cabbage, return alone, take the goat. Seven crossings, and nothing gets eaten.
Lesson 5
Lewis Carroll's The Game of Logic, published in 1886, teaches reasoning as a parlour game. The world holds Things, such as buns, babies, and beetles, and Things have Attributes, such as baked, beautiful, and broken. A Proposition says that some, none, or all of one class of Things belong to another class, as in some new cakes are nice. The class talked about is the Subject, the class it is matched with is the Predicate, and the whole set of Things under discussion is the Universe of discourse, so Carroll cheerfully begins: let us take a Universe of cakes.
A syllogism is two Propositions, the Premisses, from which a third, the Conclusion, can be proved. One Attribute appears in both Premisses, forming the Middle Term, and it disappears in the Conclusion. Here is Carroll's first example, worked in words. Some new cakes are unwholesome. No nice cakes are unwholesome. Consider those new cakes that are unwholesome: they cannot be nice, because no nice cake is unwholesome. So some new cakes are not nice, and that is the Conclusion. His second example is grander: all dragons are uncanny, and all Scotchmen are canny, so all dragons are not Scotchmen and all Scotchmen are not dragons. Note that Carroll never promises the Premisses are true. Logic tests only whether the Conclusion follows if they are.
Lesson 6
Carroll tests syllogisms on diagrams, not feelings. The smaller, biliteral diagram is a square cupboard for one Universe. The top half holds Things with attribute x, the bottom half the not-x Things, the left half y, and the right half not-y, so each compartment carries two attributes, written on its edges. A red counter (written 1) means a compartment is occupied; a grey counter (0) means it is empty. If you know only that some x Things exist, not whether they are y or not-y, the red counter sits on the fence, on the dividing line. All x are y packs two facts together: some x are y, and no x are not-y.
The larger, triliteral diagram adds a third attribute m, a square inside the square, so each compartment carries three attributes, read from its corners. Carroll's method: pick your Universe, let m be the middle attribute, and mark the negative Premiss first, because grey counters are certain. In the cakes problem, no nice cakes are unwholesome empties two compartments at once. Then some new cakes are unwholesome wants a red counter in one of two compartments, the grey counter has already emptied one, and the red counter is forced into the other. Whatever transfers back to the small diagram is your Conclusion; if nothing transfers, there is no conclusion, however fine the argument sounds. His trap proves it: all soldiers are brave and some Englishmen are brave yield no conclusion at all.
Lesson 7
Now build one. Dudeney's advice, gathered from his dissection chapters, makes a fine recipe. Keep the conditions simple enough to say in one breath, even if the solution is difficult, because nobody attempts a puzzle they cannot understand. Solve your own puzzle completely before showing it to anyone, and write the solution down, since a puzzle with no known answer is just a rumour. Aim for the fewest pieces, the fewest moves, or the neatest condition you can manage, because elegance is what makes a puzzle worth repeating.
Learn his favourite improving trick: add one condition to kill spare solutions. Cutting a Greek cross into four pieces to form a square can be done in infinitely many ways, so alone it is a weak puzzle. Require all four pieces to be the same size and shape, and there is only one solution, and it is beautiful. Respect exactness, too. Dudeney mocks a boy who thought a near cross was near enough, noting the boy would not have accepted fourpence change from sixpence for a penny apple. Finally, test your puzzle on a friend. If they stumble on your wording, rewrite the conditions. If they stumble on the idea, your puzzle is working. Keep a dated notebook of puzzles and solutions.
Sources
This course is taught from classic texts that are free for anyone to read. Every method in the lessons traces back to one of these books.
Graduation
You started with a magic square and finished with Carroll's counters in your hands and a puzzle of your own in your notebook. Keep solving slowly, keep checking every answer against the conditions, and keep your own solutions written down. The next locked door you meet, in a book or in ordinary life, should feel a little less locked. Congratulations, and happy puzzling.
Keep learning
This course is part of Starvard University, striving for greatness.
A note on accuracy: Much of what you read at Starvard was created with help from AI agents, working alongside people. We check what we can, but we cannot guarantee that everything here is true, current, or complete. If something really matters, your health, your money, a grade, a job, please verify it with a trusted source before you rely on it.