Browsing category arithmetic

Binet’s formula and Haskell

This is an extended version of my entry in the Lockdown Mathoff at the Aperiodical Binet’s formula1 is a lovely way to generate the $n$th Fibonacci number, $F_n$. If $\phi = \frac{1}{2}\left(\sqrt{5} + 1\right)$, then $$F_n = \frac{ \phi^n - (-\phi)^{-n} }{\sqrt{5}}$$ Haskell and computation The main reason I’m writing

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An excellent puzzle I heard from @panlepan (I paraphrase, as I've lost the tweet): When you move the final digit of 142857 to the front, you get 714285, which is five times as large. What is the smallest positive integer that is doubled when the last digit moves to the

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Back in 1940…

That @solvemymaths is an excellent source of puzzles and whathaveyou: Meanwhile, back in 1940 when everything was basically shit... — Ed Southall (@solvemymaths) October 7, 2017 How would you find $\sqrt[3]{\frac{1-x^2}{x}}$ when $x=0.962$, using log tables or otherwise? I would start by trying to make the numbers nicer: I

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Long division

Some time back in the olden days, @robeastaway posted this: My 11 year old's sample SATs papers found in his book bag. "Work out 1118÷43 (no calculator)." I wish all ministers had to take these tests. — Rob Eastaway (@robeastaway) March 22, 2017 Before I say anything else: I

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Ask Uncle Colin: fourth roots

Dear Uncle Colin, How would you find $\sqrt[4]{923521}$ without a calculator? -- Some Quite Recherché Technique Hi, SQRT! I have a few possible techniques here. The first is "do some clever stuff with logarithms", the second is "do some clever stuff with known squares" and the last is "do some

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The Mathematical Ninja lets the student investigate… cube roots

"Sensei! I have a problem!" The Mathematical Ninja nodded. "Bring it on." "There's a challenge! Someone has picked a five-digit integer and cubed it to get 6,996,364,932,376. I know it ends with a six, and I could probably get the penultimate digit with a bit of work... I just wondered

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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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Ask Uncle Colin: How do I multiply big numbers together?

Dear Uncle Colin, How do you multiply big numbers like $2158 \times 1812$? I try to do it using the column method or the grid, but I always make mistakes. -- A Desperately Desired Error Reduction Hi, ADDER, and thanks for your message! I've been playing with something midway between

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On recurring decimals

It's encouraging to see a few less-predictable questions coming up in the new GCSE and A-level specifications. @mathsjem highlighted an especially nice GCSE one: Question 26 from yesterday's Edexcel Methods 2 GCSE paper. Helpful for revising recurring decimals. — Jo Morgan (@mathsjem) June 17, 2016 This is unusual more

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Brutal simultaneous equations

I recently became aware of the IYGB papers, available from Madas Maths. Like the Solomon papers, they're intended to stretch you a bit -- they're ranked by difficulty from standard to extremely hard. My student, being my student, demanded we go through one of the extremely hard ones. There were

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