Time for another story? (Thread)
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One of the puzzles in that book (originally published in Strand Magazine in 1902) concerned Lady Isabel’s Casket.pic.twitter.com/wREbOdtMxN
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In 1935, a young man called Arthur Stone went to Trinity college, Cambridge, to study mathematics. He had a copy of the Canterbury Puzzles, and he was particularly intrigued by Dudeney’s assertion that “This is the only possible solution”.
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It occurred to him that this claim of uniqueness had some profound implications. If a square could be divided into smaller squares, all of different sizes, then another solution to the casket problem could be obtained by subdividing the smallest square.
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So, if Dudeney was right that the solution to the casket problem was unique, it must therefore be impossible to cut a square into smaller squares of different sizes. Stone wondered whether this was true.
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He discussed the question with his friends Cedric Smith, Leonard Brooks, and Bill Tutte. The others were intrigued, and began to work on it together.
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They started by thinking about how to dissect rectangles into squares of different sizes. They knew that was sometimes possible, because Tutte had a copy of a book by Rouse Ball which showed how a 32×33 rectangle could be cut into different-sized squares.pic.twitter.com/06c9u1Fmkg
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They came up with a clever method for finding squared rectangles. First divide a square into rectangles willy-nilly; then pretend the rectangles are really squares, and work out how big they would be if they were, and solve a simple equation to find the dimensions.pic.twitter.com/SnPzq5YEJZ
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In this example, you start with arbitrary values x and y, fill in the rest of the rectangles starting from those, and then finally you have to make the lengths match on the line AB, so solve the equation 14y-3x = (3x-3y) + (3x+y). It simplifies to 16y = 9x, so x=16, y=9 works.
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The big breakthrough came when Smith invented a type of diagram that led the four to realise that squared rectangles are equivalent to electrical circuits – specifically networks of resistors.pic.twitter.com/1DNf7y8RXe
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It works like this: think of the squares as 1Ω resistors, and as electrically connected whenever they share a horizontal line segment. Connect a battery to the top and bottom of the rectangle.
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Think of the current flowing through each resistor as the side-length of the square. The current that flows into each node must equal the current that flows out, which corresponds to the fact that the squares above and below a shared horizontal segment have the same total width.
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This correspondence meant the students could use Kirchhoff’s theory of electrical networks to study squared rectangles, and they found many more. One was this, which impressed Brooks so much that he drew it on cardboard and cut it up to make a puzzle.pic.twitter.com/MhQpRxUs0G
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Home for vacation, he challenged his mother to solve the puzzle. He was astonished when she found an entirely unexpected solution.pic.twitter.com/VON9lGHRlv
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The corresponding circuits are here. The two points marked p and p’ are at the same voltage, so they can be connected together without changing the electrical behaviour of the circuit.pic.twitter.com/QO19uXNfu6
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Studying this phenomenon led Tutte to an idea he called “rotor-stator equivalence”. Using this idea, Smith and Stone managed to find a perfect square made of 69 smaller squares. They rushed to tell Brooks, who replied “So have I!”. He had found one too, also of 69 squares.
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Their work was published in 1940 under the title “The dissection of rectangles into squares”. They didn’t quite manage to publish the first perfect squared square: they were beaten to press by R Sprague, who had found one independently and published in 1939.
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But their work led to something of far greater importance. The Second World War broke out in 1939, and Bill Tutte’s tutor recommended him for war work at the Government Code and Cypher School at Bletchley Park.
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Tutte was officially studying chemistry, and presumably was only recommended for Bletchley Park because his work on squared squares had shown him to be a talented mathematician.
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At Bletchley Park, Tutte had the key insight that led to the cracking of the Lorenz cipher, used by the Wehrmacht for their most secret communications. The intelligence derived from these communications may have made the allied victory possible. It definitely helped.
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So, in a roundabout way, a puzzle about a Lady’s casket led to the defeat of Hitler. Remember that next time someone says puzzles are a waste of time.
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Arthur Stone went to Princeton, where he almost immediately discovered flexagons. https://en.m.wikipedia.org/wiki/Flexagon
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The website http://squaring.net has a HUGE amount of information about cutting rectangles into smaller squares, including a detailed history section. Tutte’s book _Graph theory as I have known it_ tells the story from Tutte’s perspective, with an emphasis on the mathematics.
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Smith’s paper _Did Erdős Save Western Civilization_ describes what happened from Smith’s point of view. (Erdős conjectured that it’s impossible to cut a square into smaller squares of different sizes, though the four friends weren’t aware of that at the time.)
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Here’s the most efficient possible way to cut a square into smaller squares of different sizes, discovered by Duijvestijn in 1978.pic.twitter.com/WZSck6Ox8E
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If you enjoyed the story above, you might enjoy this documentary about Bill Tutte’s work at Bletchley Park breaking the Lorenz cipher: …https://computer-literacy-project.pilots.bbcconnectedstudio.co.uk/7fd3fb55e462db0867b183729c5ed27c … Via
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