Posted in ask uncle colin.

Dear Uncle Colin, I need to figure out $\int \cos^3(2t) \sin^5(2t) \dt$ and I’m… just going round in circles. So to speak. What do you suggest? - Doing Integration’s Really A Chore Hi, DIRAC, and thanks for your message! It’s very easy to end up going around in circles on

Read More →
Posted in puzzles.

A nice puzzle by way of @benjaminleis: This AIME problem is fun: pic.twitter.com/DhbviTqnqr — Benjamin Leis (@benjamin_leis) February 2, 2020 In case you can’t read that, we need to find the sum of the digits in $N = 9 + 99 + 999 + 999\dots999$, where the last number consists

Read More →
Posted in ask uncle colin.

Dear Uncle Colin, I want to stretch my Year 12 Further Maths class - what extra-curricular topics would you recommend? Something To Really Engage Their Creative Reasoning Hello, STRETCH, and thanks for your message! How excellent to be looking beyond the curriculum for ways to engage and develop your young

Read More →
Posted in number theory.

I’m in the process of clearing out old bookmarks, and stumbled on this puzzle from @jase_jwanner: Prime or not prime? No calculators allowed!a. 23567897614^2 - 1b. 34564344^3 -1c. 76543556556625731d. 345643554^{10} - 169 — Jase (@jase_jwanner) August 27, 2016 I shall give you a moment to ponder these, and put my

Read More →
Posted in ask uncle colin.

Dear Uncle Colin, I’ve figured out that $x^{x^{x^{\dots}}} = 2$ when $x = \sqrt{2}$, but I’m struggling to make sense of the function - it seems to have a vertical gradient when $x = e^{\frac{1}{e}}$, but it doesn’t seem to have what I think of as an asymptote there. What

Read More →
Posted in ninja maths.

“We’ve been through this a hundred times, sensei. I say something like ‘$10^{1.35}$. Hm, let me get my calculator’ and you torture me in some unspeakable way an blurt out the answer…” “22.4” “… thank you, especially for refraining from the torture bit.” “You’re welcome.” “Then, of course, you tell

Read More →
Posted in ask uncle colin.

Dear Uncle Colin, On a recent revision course, my tutor couldn’t integrate $\int_0^{\piby2} \frac{\sin(2\theta)}{1+\cos(\theta)} \d \theta$. Can you? - Reasonable Expectation For University’s Nameless Don? Hi, REFUND! Far be it from me to disparage a fellow professional’s integration skills, especially in the heat of the moment; I frequently mistake $2\times

Read More →
Posted in dome.

I’m a big fan of the doodle. My lecture notes, even my schoolbooks, are covered with geometric patterns and impossible shapes and simple cartoons. Today’s entry in the Dictionary of Mathematical Eponymy started jumped out of Stanislaw Ulam’s notes while he was listening (according to Martin Gardner) to a ‘long

Read More →
Posted in ask uncle colin.

Dear Uncle Colin I wonder: at what height is the volume of a cone above that height equal to the volume below? What about the surface area? Are there any cones where it’s the same height? Finally Researching Uniformly Splitting Things Up, Mate Hi, FRUSTUM, and thanks for your message!

Read More →