Browsing category proof

A proof without words

Via nRICH: A circle touches the lines OA extended, OB extended and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle. $\blacksquare$

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A question that comes up a lot in class is, “how do you get good at proofs?” (It’s usually framed as “I don’t like proofs”, but we’re not having any of that negativity here, thank you very much.) I don’t have a silver bullet for that. I do have some

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A strange number base

Aaaages ago, @vingaints tweeted: This is pretty wild. It feels like what the Basis Representation Theorem is for Integers but for Rational Numbers. Hmm - trying to prove it now. Feels like a tough one. Need to work some examples! — Ving Aints (@vingAints) September 18, 2018 In

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Ask Uncle Colin: A Trigonometric Proof

Dear Uncle Colin, I have a trig identity I can't prove! I have to show that $\frac{\cos(x)}{1-\sin(x)} = \tan(x) + \sec(x)$. Strangely Excited Comment About Non-Euclidean Trigonometry. Hi, SECANT, and thanks for your message! This is a slightly sneaky one, but definitely a good one to practice. Let's do it

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Several Strings of 1s

This puzzle was in February's MathsJam Shout, contributed by the Antwerp MathsJam. Visit to find your nearest event! Consider the set ${1, 11, 111, ...}$ with 2017 elements. Show that at least one of the elements is a multiple of 2017. The Shout describes this one as tough; you

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Ask Uncle Colin: A Cosec Proof

Dear Uncle Colin I'm stuck on a trigonometry proof: I need to show that $\cosec(x) - \sin(x) \ge 0$ for $0 < x < \pi$. How would you go about it? - Coming Out Short of Expected Conclusion Hi, COSEC, and thank you for your message! As is so often

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How Would Martin Gardner Prove It?

Someone recently asked me where I get enough ideas for blog posts that I can keep up such a 'prolific' schedule. (Two posts a week? Prolific? If you say so.) The answer is straightforward: Twitter Reddit One reliable source of interesting stuff is @WWMGT - What Would Martin Gardner Tweet?

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A Digital Root Puzzle

Every so often, a puzzle comes along and is just right for its time. Not so hard that you waste hours on it, but not so easy that it pops out straight away. I heard this from Simon at Big MathsJam last year and thought it'd be a good one

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From Euclid to Cantor

One of my favourite quotes is from Stefan Banach: "A good mathematician sees analogies between theorems. A great mathematician sees analogies between analogies." This post is clearly in the former camp. I'm fairly sure it's a trivial thing, but it's not something I'd noticed before. One of the first serious

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Repdigit endings to squares

Over at @onthisdayinmath, Pat highlights a @jamestanton question about squares: $2^2$ ends with 4 and $12^2$ ends with 44. Is there a square than ends 444? How about one that ends 4444? Pat's answer (yes to the first -- $38^2 = 1444$ is the smallest -- and probably not to

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