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?
Pythagoria is a puzzle game for PCs. It’s the same idea as Naoki Inaba’s Area Maze: you’re shown a geometric construction, not drawn to scale, and you have to work out a missing length or an area.
Each puzzle is constructed so that it can be solved without ever dealing with fractions, though what exactly that means is up for debate. Whatever it means, it keeps you from breaking out pen and paper to solve a problem algebraically, when you know there should be a way of doing it in your head.
My wife’s grandmother is a fearsome character. She’s in her nineties but still has all her wits about her. In fact, she’s got more than her fair share of wits. Whenever we visit her, she hits me with a barrage of questions and puzzles collected from the last several decades of TV quiz shows and newspaper games pages. My worth as a grandson-in-law is directly proportional to how many answers I get right.
One of her favourite modes of attack is the “30 Second Challenge” from the Daily Mail. It looks like this:
You start with the number on the left, then follow the instructions reading right until you get to the answer at the end. It’s one of Grandma’s favourites because it’s very hard to do in your head when she’s just reading it out!
Sam’s dad is in a mathematical conundrum – so she’s asked Katie, one of our editors, if maths can save the day.
Dear The Aperiodical,
My dad is going away on a golfing holiday with seven of his friends and, since I know a little bit about mathematics, he’s asked me to help him work out the best way to arrange the teams for the week. I’ve tried to work out a solution, but can’t seem to find one that fits.
They’ll be playing 5 games during the week, on 5 different days, and they’d like to split the group of 8 people into two teams of four each day. The problem is, they’d each like to play with each of their friends roughly the same amount – so each golfer should be on the same team as each other golfer at least twice, but no more than three times.