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A visual proof that a^2 – b^2 = (a + b)(a – b)
A visual proof that a2 – b2 = (a + b)(a – b). Sophie Germain wrote, “It has been said that algebra is but written geometry and geometry is but diagrammatic algebra.”
Greg Ross
Futility Closet is a collection of entertaining curiosities in history, literature, language, art, philosophy, and mathematics, designed to help you waste time as enjoyably as possible.
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1/999999999999999999999998999999999999999999999999
Divide the number 999,999,999,999,999,999,999,998,999,999,999,999,999,999,999,999 into 1 and express the result as a decimal expansion, and you’ll find the Fibonacci sequence presented in tidy 24-digit strings: (Thanks, David.)
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Images Created from Prime Numbers
Inspired by James McKee’s Trinity Hall prime, physics researcher Gilles Esposito-Farese (of the self-descriptive pangram) has worked out that this 2,258-digit prime number: 53084469288402965415857953902888840109250315594591772197851275320 59910735745658243457138160802170063601085186072703319516241231606…
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Albert Einstein in 1939 composed a message for the people of AD 6939
Preparing a time capsule in 1939, the Westinghouse Electric & Manufacturing Company asked Albert Einstein to compose a message for the people of AD 6939. He sent this: Our time is rich in inventive minds, the inventions of which could facilitate our lives considerably. We are crossing the seas by power and utilize…
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Einstein went to his office just so he could walk home with Gödel
Albert Einstein used to say that he went to his office at the Institute for Advanced Study “just to have the privilege of walking home with Kurt Gödel.” The two would meet at Einstein’s home each day between 10 and 11 and undertake the half-hour walk to the institute. At 1 or 2 in the afternoon they’d walk back…
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Perfect example of real-life hacking
In 1999, UC-Davis civil engineer David Phillips was grocery shopping when he noticed something peculiar. Healthy Choice Foods was offering frequent-flyer miles to customers who bought its products. But a 25-cent pudding would bring 100 miles — the reward was worth more than the product itself. Recognizing a good…
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Lewis Carroll – computing the day of the week for any given date (1887)
A letter from Lewis Carroll to Nature, March 31, 1887: Having hit upon the following method of mentally computing the day of the week for any given date, I send it you in the hope that it may interest some of your readers. I am not a rapid computer myself, and as I find my average time for doing any such question is…
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Topologist's Map of the World
Reddit user xilefakamot created this in 2020 — a “topologist’s map of the world.” It depicts international borders and nothing else — the shapes, sizes, and distances between countries are forsaken in order to present the pattern of their borders as efficiently as possible. “No calculations per se – I sketched out a…
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New Mexico: Psychologists to dress up as wizards when providing expert testimony
New Mexico’s state senate took up a startling amendment in 1995 — it would have required psychologists to dress up as wizards when providing expert testimony on a defendant’s competency: When a psychologist or psychiatrist testifies during a defendant’s competency hearing, the psychologist or psychiatrist shall wear a…