When you discover an interesting mathematical shape or object, there’s a strong instinct to play with it – maybe by drawing sketches and doodles to test the limits of the idea. But in the case of Tricurves, drawing an accurate shape takes a little time, and it doesn’t lend itself well to idle experimentation.
Producing a physical version of a shape, in enough quantity to allow for experimentation, makes it much more tangible. In our own respective locations, we’ve each made use of laser cutting facilities to produce wooden Tricurve tiles to play with, and we encourage you to join in.
This is just a quick post to tell you about a nice puzzle game I spotted on Mathstodon.
It’s called Transposition, and it’s a sliding block puzzle in the vein of the popular game Rush Hour. You’re given a grid that’s almost full of rectangular blocks, and you have to slide them around each other until the two coloured blocks have swapped places.
The puzzle was invented by mathematician Jacob Siehler, who says he used a computer search to generate a pool of puzzles, given the rules of the game. I took quite a while to solve all 5 “easy” puzzles – as with any logic puzzle, you need to play about for a while to get a feel for the mechanics. I hadn’t appreciated at first that the grey blocks don’t need to be in their starting places when you solve the board – only the coloured blocks need to in the right positions.
There are 26 puzzles at the moment, ranging from “easy” to “very hard”. Have a crack at it! I really enjoyed it.
A few months ago, my faculty’s PR person sent an email round asking if anyone would like to write a puzzle for the Today programme’s “Puzzle for Today” slot, to be broadcast during the programme’s trip to Newcastle in Freshers’ Week. A colleague said this might be the kind of thing I’d like to do, which it was, so I started thinking, and eventually came up with a brand new puzzle which I thought would work well.
If you listened to the Today programme this Thursday morning, you’ll have heard not my name, but that of Dr Steve Humble, who’s got a lot more experience doing this kind of thing. Turns out, they wanted something more ‘visual and interactive’, so asked him instead. I think that was a polite way of saying they just didn’t like my puzzle. Oh well!
Steve chose a classic puzzle that coincidentally appeared on Twitter about a month ago, prompting much discussion. It’s a good puzzle, much better than the one I came up with, but I don’t think Steve was completely right to say “It is possible that you can always create a winning game” – that’s only the case if there are an even number of coins, but his statement said “around ten coins”. I suppose he might’ve meant that, starting from having a handful of coins, you can decide to only use an even number of them.
The upside is that I can now talk about the puzzle here, where someone might actually enjoy it.
Longtime friend of the Aperiodical, artist, mathematician and #BigMathOff semifinalist Edmund Harriss has come up with a new puzzle/toy/exploration set, developing his Curvahedra system. We asked him to explain the maths behind it in this guest post.
Curvahedra is a flexible system of connectors that can make all sorts of different things, combining puzzles (and self-created puzzles) with art. You can get your own to play with, explore, prepare for Christmas (they make great decorations, wreaths and presents) at our online store, and get 15% off with the discount code APERIODICAL.
As this is the Aperiodical, you might be most interested in how it can be used to explore mathematics. In the big math off I talked about the basic ideas behind the system, Gauss’ famous Theorema Egregium and Gauss-Bonnet theorems. A really simple version of this comes from just considering triangles, that can be built up to make this:
Author and mathematician Josh Holden has come up with a nice puzzle — so we’re posting it as a competition. If you think you can decrypt the message below, send in the decrypted message and a one- or two-sentence description of the mathematical principle behind the encryption key to firstname.lastname@example.org. The first correct entry received will win a copy of Josh’s book, The Mathematics of Secrets.
The technical name for the “cryptograms” found in many newspapers and magazines is monoalphabetic monographic substitution ciphers — monographic meaning that they make substitutions one letter at a time and monoalphabetic meaning that the substitution rule is the same every time a given letter appears in the message.
Most often the easiest way to start solving these is to look at one-letter words which are usually “I” or “a”, then two-letter words, etc. If the breaks between the words are removed, then you might use the fact that in a typical English text the letter “e” will occur about 13% of the time, followed by “t” and “a” at 7-8% and others farther behind.
What then should we make of the following cryptogram?
The letter “Q” appears almost 20% of the time, followed by “F” at about 10%, and “Y” and “T” at about 8%. The original text is English (in fact it’s from a famous work of children’s literature) and it doesn’t have a particularly odd distribution of letters. Can you decrypt the message? For bonus points, can you figure out what is mathematically interesting about the encryption key?