C: $K_A m; \\ K_B d.$
A: $\neg K_A d; \\ m \vDash \neg K_B m.$
B: $d \not\vDash K_B m; \\ (K_A(\neg K_B m)) \vDash K_B (m,d).$
A: $m \wedge K_B(m,d) \vDash K_A (m,d).$
Albert, Bernard and Cheryl have had a busy week. They’re the stars of #thatlogicproblem, a question from a Singapore maths test that was posted to Facebook by a TV presenter and quickly sent the internet deduction-crazy.
First of all: no, it’s not meant to be answered by an average Singaporean student. It’s a hard question from a schools Olympiad test.
Happy π day everyone! I hope you’re having a great day, and having lots of fun mathematical parties.
You may have noticed that here at The Aperiodical, we’ve been posting exciting π-related items all week – and here’s a list of them all, collected into one handy place. Enjoy!
For a while, I’ve been following this cool Twitter account that tweets questions Wolfram|Alpha can’t answer. The genius of it is that the questions all look like things that you could half-imagine the solution algorithm for at a glance, and many of them look like the kinds of questions Wolfram like to give as examples when they’re showing off how clever their system is.
Questions like this:
The answer to that is 278. How do I know that? I know that because I went on a little problem-solving binge answering the questions that Wolfram|Alpha can’t.
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.
Can you help me figure it out?
Sam Coates, Manchester
Bees have encouraged mathematical speculation for two millennia, since classical scholars tried to explain the geometrically appealing shape of honeycombs. How do bees tackle complex problems that humans would express mathematically? In this series we’ll explore three situations where understanding the maths could help explain the uncanny instincts of bees.
A curvy wild honeycomb.
Honeybees collect nectar from flowers and use it to produce honey, which they then store in honeycombs made of beeswax (in turn derived from honey). A question that has puzzled many inquiring minds across the ages is: why are honeycombs made of hexagonal cells?
The Roman scholar Varro, in his 1st century BC book-long poem De Agri Cultura (“On Agriculture”), briefly states
“Does not the chamber in the comb have six angles, the same number as the bee has feet? The geometricians prove that this hexagon inscribed in a circular figure encloses the greatest amount of space.”
(A report by Richard Elwes from the launch of the London Mathematical Society’s 150th birthday year. All the talks are available to watch online at the LMS’ birthday portal)
There’s a standard format for celebrating a mathematical milestone, perhaps the 80th birthday of some deeply eminent number theorist. His collaborators and graduate students, and their graduate students, and their graduate students all gather together in some gorgeous location to regale each other with their latest theorems, while the rest of the world pays no attention. For the London Mathematical Society’s birthday, we had something different. Well, we did have the gorgeous location. The Goldsmiths’ hall in London is a magnificent venue, and the livery hall in particular was evidently designed by someone with a peculiar fondness for Element 79. (See for yourself.) But speaker-wise, a decision had obviously been taken that the party would be an outward-looking affair. The focus was not so much on the LMS, or even on maths per se, but on our subject’s ability to unlock worlds, particularly the worlds of TV, film, and computer games.
In an idle moment of wondering, I asked a simple question on Twitter:
The response was overwhelming. Here’s a guide to the non-existent number crunchers you should know about, and some you probably already do.