All Squared, Number 9: Miscellanea with CP and Cushing

We have an unusual All Squared podcast for you this time. My good friend David Cushing has been asking to do a podcast for absolutely ages. We couldn’t decide on a single topic to talk about, so instead I suggested we just sit down and chat about maths in general, like we do when there isn’t a microphone in front of us.

We talked for about an hour and a half, but because I’m completely stupid we lost a big chunk of it when the microphone switched off. To make things even worse, we recorded in a room with a ridiculously loud fan, so there’s that to contend with. Anyway, we talked about some fun stuff, so I think it’s worth listening to.

Here are some links relevant to the things we talked about.

• David would like you to know that $5 \times 16017 = 80085$.
• The book David brought was Topology, by James Munkres.
• Brouwer’s fixed-point theorem says that for any continuous function $f$ with certain properties mapping a compact convex set into itself there is a point $x_0$ such that $f(x_0) = x_0$.
• The pancake theorem is referred to by MathWorld as “a two-dimensional version of the ham sandwich theorem”, so CP wins. The ham sandwich theorem says that the volumes of any $n$ $n$-dimensional solids can be simultaneously bisected by an $(n-1)$-dimensional hyperplane.
• The hairy ball theorem says that there is no nonvanishing continuous tangent vector field on even-dimensional $n$-spheres – there’s always a point on the sphere where the function is zero.
• The black hole information paradox says that it’s possible for a black hole to destroy information. In 2004 Stephen Hawking conceded his bet that information is destroyed, so CP wins again. (Guess who’s writing this summary)
• The book CP brought was Only Problems, Not Solutions by Florentin Smarandache. It turns out he’s a bit of a character!
• The family of sequences which contains a sequence for each digit, except inexplicably 1, was “Primes with $n$ consecutive digits beginning with the digit $D$”
• Problem 102 of Only Problems… contained this cool diagram:
• We can’t remember what “the Russian book” was. Sorry!
• The powerful numbers are sequence A060355 in the OEIS.
• Paul Erdős made a conjecture on arithmetic progressions.
• The Bee Gees consisted of brothers Barry, Robin, and Maurice Gibb. That’s three people: a powerful triple.
• $x^2 – 8y^2 = 1$ is a Pell equation, and the reason why the continued fraction representation of $\sqrt{8}$ generates consecutive pairs of powerful numbers.
• The Muddy Children puzzle i