Season 6 Episode 6

In this episode, we've got lots of left-over ideas and bits and pieces that you might like to look into yourself over the summer.

Sequences

The On-Line Encyclopedia of Integer Sequences is at https://oeis.org/. It's very helpful if you've got a bit of a sequence and you want to see if it matches any other famous (or not-so-famous) sequences.

For an encyclopedia entry on the Thue-Morse sequence, including some strange infinite products, see Wolfram MathWorld. Wikipedia contains slightly different details, and includes a chess application. The idea of generating the sequence as a tree comes from L-systems.

The ruler sequence / surprise sequence / number-of-times-2-divides-a-number sequence is on OEIS here.

I've remembered that OEIS has a feature to turn sequences into MIDI files; https://oeis.org/play. Maybe this is how someone in chat made a MIDI file so quickly?!

57 dice

See Wikipedia for some (quite advanced) description of the "university" method to count these cubes.

Odd neighbours

I didn’t find the BIO problem that someone mentioned in chat, but I did enjoy going down the rabbit-hole of past BIO questions. See https://www.olympiad.org.uk/ if you’re interested in solving problems with computers.

Truchet Tiles

See Season 5 Episode 9 for more Truchet Tile fun

Sphinx Tiling

For the answer to "how, precisely, do you draw the Sphinx tile", see this page and then check out the fantastic page on rep-tiles.

Solving differential equations

To learn more about solving differential equations see, for example, this guide. To follow this, you’ll need to have seen implicit differentiation, the product rule, and the chain rule. That page has links learn about those too, though!

Other resources

Here are some other things to fill your time over the summer.

I really like NRICH's Complex Adventure for learning about complex numbers.
The STEP database has hundreds of hard maths problems. Searching for "C1" gives problems loosely based on the first bit of A-level Maths.
STEP Support Programme helps you to solve those hard maths problems.
Parallel Circles for more livestreams (but at time of writing there’s nothing scheduled for ages 15+; tell a younger sibling about these!).

If you want to get in touch with us about any of the mathematics in the video or the further reading, feel free to email us on oomc [at] maths.ox.ac.uk.