US organisations the Mathematical Sciences Research Institute (MSRI) and the Children’s Book Council (CBC) have founded a youth book prize, called *Mathical: Books for Kids from Tots to Teens**.* The prizes, awarded for the first time this year, recognise the most inspiring maths-related fiction and nonfiction books aimed at young people. This year, they’ve awarded a set of prizes for books released in 2014, as well as honouring books published been 2009 and 2014, plus two ‘hall of fame’ winners from the further past.

The selection committee was chaired by maths author Jordan Ellenberg, mathematical sciences professor Rebecca Goldin, and young people’s literature ambassador Jon Scieszka. The panel comprised maths specialists, teachers, computer science professors, and chairs of organisations related to maths and young people.

The awards were classified into pre-K (under 4), grades K-2 (roughly UK KS1), grades 3-5 (KS2), grades 6-8 (KS3) and grades 9-12 (KS4/5). The list of winning books, and more information about the award, are available on the Mathical Books website.

Mathical Books: Award Winners Announced

]]>**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.

You’ve doubtless seen multifarious debates about the correct answer on your Facebook/Twitter/Friendface feeds. The various newspapers of the world saw an opportunity for some easy page views and each wrote basically the same story: *When is Cheryl’s birthday? Only maths weirdos can possibly work it out.*

By far the best treatment, in my opinion, is by top pop maths chap Alex Bellos in the Guardian. He’s written a few pieces: first asking “can you solve” (with an encouraging number of fallacious arguments for each answer in the comments) and then explaining “how to solve” with his model solution. Later on, Alex was invited onto BBC TV to present his solution.

Over on Radio 4, the *Today* programme got carried away with logic puzzles and presenter Sarah Montague managed to read out the wrong answer to a hats puzzle.

**STOP PRESS: **while this post was working its way through our editing process, Numberphile posted its own explanation of the problem.

Following a discussion on Facebook where Phil Walker of Leeds University gave a firmer epistemic reason for the mainly-incorrect answer of August 17, Alex invited James Grime to write about this alternate line of reasoning. I’m still not having it though. And because everything written on the internet has to subsequently become a video, James has posted his explanation of the different interpretations to his YouTube channel.

If you’re still unsure, or still being bombarded by relatives who want an explanation, have a look at this fantastic interactive explanation by Mark Josef. When you click on the day you think is Cheryl’s birthday, it walks through the conversation (as well as Albert and Bernard’s internal monologues) and highlights any statements that would be inconsistent.

By the way, my wife^{1} asked why the characters in the original puzzle were called Albert, Bernard and Cheryl. In case you’re also wondering that, it’s a convention in this kind of thing to give the characters names whose first letters work through the alphabet, so in your working-out you can abbreviate them to A,B,C,… Famously, in cryptography problems Alice and Bob have a whale of a time sending messages to each other.

Now, with the original question definitively dealt with and the populace at large going mad for logic problems, the world’s mathmos are scrabbling to show off their favourite deduction problems.

Kit Yates is jumping on the bandwagon with this version, which he says was used as an interview question for several years:

@alexbellos @jamesgrime While people are in mood #thatlogicproblem Lets try to send another one viral #neighboursprob pic.twitter.com/OKtOK1lXBn

— Kit Yates (@Kit_Yates_Maths) April 16, 2015

Tanya Khovanova is the master of this sort of thing. Conway’s wizards on a bus puzzle is the apotheosis of the genre and Tanya has written a great paper about its solution, along with a generalised version. Probably inspired by the current hullabaloo (or maybe not – she is usually thinking about these things), Tanya posted an extremely concise puzzle to her blog yesterday.

Deduction is normally done by prisoners either in hats or with boxes. If you really want to blow your brain sockets, try the blue-eyed islanders puzzle. Terry Tao posted it to his blog once and a gargantuan comments thread full of people not getting it ensued.

Meanwhile, Tim Gowers is all “I hear you like deduction, so I put some deduction in your deduction so you can deduce while you deduce” with this transfinite version. Although Joel David Hamkins turned up in the comments and shared his version, which looks much nicer to me. Later on, he added yet another transfinite epistemic logic puzzle.

With all this talk of deduction, you might remember the QI-worthy observation that Sherlock Holmes doesn’t do deduction – he does abduction (hey, whaddya know: I did learn that on QI!). *Deductive* reasoning is the process of working from a set of premises to a certain conclusion by fixed rules of inference. *Inductive* reasoning involves using premises which provide evidence for a conclusion which is *probably* true. Finally, *abductive* reasoning is when you try to find the simplest explanation for an observation.

“CP”, you say, “that’s all well and good, but can you put all this in joke form?”

Yes.

Since this is a maths site, I should think about what all these puzzles have in common and how to formalise them. Clearly they’re logic puzzles, but the deductions based on announcements can’t be written down in classical logic. It turns out there’s a thing called public announcement logic which provides a whole load of new modal operators such as “I know $\phi$”, and “$\phi$ is not refutable”.

My go-to resource for explaining public announcement logic is *The Muddy Children:*

*A logic for public announcement*, a set of slides by Jesse Hughes.

There’s a whole load of notation around public announcement logic, involving things like Kripke frames. The whole thing got going with Jan Plaza’s paper “logics of public communications”. For a paper introducing a new field of logic, it’s surprisingly readable!

The notation at the top of this post is a mangled mash of various things that sort of represent the conversation in #thatlogicproblem but no self-respecting logician would recognise it.

- I got married on Sunday! Hooray!

The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.

]]>Puzzlebomb is a monthly puzzle compendium. Issue 40 of Puzzlebomb, for April 2015, can be found here:

Puzzlebomb – Issue 40 – April 2015

The solutions to Issue 40 will be posted at the same time as Issue 41.

Previous issues of Puzzlebomb, and their solutions, can be found here.

]]>On 25th March, the awards ceremony for the Abel Prize took place in Oslo, Norway. The prize, given by the Norwegian Academy of Sciences and Letters, is given for outstanding scientific work in the field of mathematics, and includes a cash prize of 6 million Norwegian Krone (about £500,000). This year the award went to American mathematicians John F Nash, Jr (yes, that John Nash) and Louis Nirenberg, “for striking and seminal contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis”.

Abel Prize website

2015 Prize announcement

This is slightly old, but we found it in the news pile – a study conducted in rural Guatemala found that humans, even without formal training in probability, were able to predict which of two events were more likely, even when complicated conditional probability was involved.

Humans have innate grasp of probability, at Nature

You may be familiar with the 1884 satirical novella by Edwin A Abbott, set in a two-dimensional universe and providing both social commentary and a wonderful analogy for thinking about three and four-dimensional space. Earlier this month, a team of visually impaired performance artists created an interactive experience, in which the 2D universe was simulated by removing visual information.

The installation took place entirely in the dark, and haptic feedback via robotic devices allowed visitors to explore the space, using radio signals and wifi to control a vibrating cube held by participants. An invited audience of 100 took part in the exhibit, and in groups of four, wearing special suits and interacting with live actors and recorded sounds, were told the story of the book.

The previews, described by one attendee as ‘truly disorienting’, took place over a week in March, with a view to a full public performance in 2018.

Flatland: An Adventure in Many Dimensions website

London theatre company develops cube that helps blind feel magic of the stage, at the Evening Standard

via Tarim on Twitter

In an ongoing ‘at it again’-based saga, Dr Eugenia Cheng has come up with yet another mathematical media formula, this time for the perfect aeroplane flight. Travel website Skyscanner has hired Cheng to produce the formula, which takes into account time of day, leg room, and punctuality of the flight (and nothing else) to give a score out of 20. Legroom is measured in inches, and punctuality is (of course) out of 100.

How to book the perfect flight: Maths professor invents formula to ensure the best journey, at the Irish Mirror

]]>You’ve just bought a lovely fresh haggis quiche at your local Minus 4 shop and are planning to eat it in one sitting, in your kitchen with a friend. You’ve agreed to share it in the fairest possible way: one of you cuts and the other choses. The quiche is in the usual circular shape.

A coin is tossed—rather unnecessarily, it must be said—and it is determined that your friend gets to cut. You step out of the kitchen for a moment and upon your return discover to your horror that your friend has already done the cutting, but not as you had expected. Instead of making one simple straight cut as close to a diameter as possible, the big oaf has made four straight cuts.

Despite the fact that these four cuts neatly meet at 45 degrees at a single point, everything is off-centre. The first cut is close to being a diamater, but the others are more lop-sided. The resulting eight slices are unequal in size, and the near-diameter cut clearly doesn’t bisect the quiche either.

“What have you done, Hamish?” you cry out in exasperation. “You were only supposed to make *one* cut, then I’d get to chose which piece I wanted. I was assuming you’d be smart enough to cut it into two pieces which looked so similar you wouldn’t mind which one I took for myself. That was the whole point! But now, even if we pretend you only made one of those four cuts, they’re all so skewed that it won’t seem fair, as it would be too easy for me to get four adjacent pieces that add up to more than half the quiche.”

“Sorry,” he replies sheepishly. “I thought that just two pieces would be too big. So I went for eight smaller ones. Besides, this is how I always cut deep-fried Mars bars.”

Trying to suppress the look of disbelief that has come over your face, you force Hamish to sit down for five minutes and read the Wikipedia page on fair cake-cutting.

“Now do you get it?” you ask. “With two pieces it’s easy, but with eight pieces it’s much trickier!”

“To be honest, I got lost early on reading that webpage,” Hamish confesses. Glancing over his shoulder you can see why, as the easy two-piece case is buried deep in the article.

“I’ll try to make it up to you,” Hamish offers. “Why don’t I divide these eight pieces into two piles of four, as fairly as I can, and you get to pick whichever pile you like?”

“Hmm, I guess that would work,” you reply. Then a revelation hits you. “Actually I have a better idea. Please let *me* do the division into two piles of four pieces, and you can chose whichever pile you want. Furthermore, I’ll do this with my eyes closed. You can even rotate the plate first so I don’t even know which piece is where. And I’ll bet you a tenner that I’ll do such a good job that no matter which pile you chose you won’t get much more than me! You can verify using the kitchen scales over there. You can even change your mind after you choose and weigh. You can’t ask for more than that!

It’s Hamish’s turn to be incredulous. He can’t believe his luck, being convinced that he’ll soon be £10 richer, since he doesn’t think you’ll be able to pull it off as you claim.

You close your eyes as promised, having first placed two identical clean plates on the table, one to the left of the quiche, the other to the right. Hamish spins the plate in front of you and says, “Ready? You can touch but don’t look.” You reach forward and quickly remove four pieces of the quiche and set them on the plate on the left. You open your eyes and put the remaining four on the plate on the right. Hamish picks one plate for himself, and then weighs both of them. To his astonishment he finds that he has lost the bet fair and square, even if he switches plates at this late stage. Crestfallen, he hands over ten quid.

What is it you do that guarantees your success here? This picture reveals all.

Even though your eyes are closed, it’s easy to remove four alternate slices to the plate on the left. If the four slices left behind are placed on a similar plate, and the two are weighed, they should come out more or less equal.

This works because of the little known and oft-misnamed Quiche Theorem, which asserts:

If a circular quiche is divided into

4nslices (wherenis at least 2), by making cuts at equal angles through an arbitrary internal point, then the sums of the areas of alternate slices are equal.

Okay, so that’s really a 2D result about area, but we’ve just added depth (and flavour) to it.

This applies when we have 8, 12, or 16 slices, but not if there are 4, 6, 10 or 14. Furthermore, since the crust on a quiche can be viewed as the difference between the whole quiche and a slightly smaller circle of filling, with the same centre, and each circle behaves according to the theorem, it follows that you and Hamish also get equal amounts of crust.

Curiously, in cases where the number of slices isn’t a multiple of 4, the alternate slice strategy gives one person more quiche but less crust, and vice versa.

For 8 slices, Carter & Wagon (1994) gave a disection proof of the alternating slice strategy.

Maybe you should let Hamish loose with the knife a little longer, then he could see for himself, without resorting to weighing, that the quiche can be split into two equal-weight halves consistent with his initial 4 cuts. He’d then have sixteen mostly bite-sized pieces, half of which would match up with the other half, down to the last crust.

THE END

*If you need to see the theorem in action, here’s a reproduction of Carter and Wagon’s proof without words in a GeoGebra worksheet. Drag the big blue points around to change the slices.*

Ooh, I get to break out my “holy power law, Batman” image again! Yippee!

Ctrl+F “power law” – no hits. That’s odd.

Follow the link to the story “Ancient cities grew pretty much like modern ones, say scientists” in the Christian Science Monitor.

Ctrl+F “power law” – no hits. Hm! What could this mysterious mathematical rule be?

Follow the link to the research group’s website. Oh look, it’s the same Geoffrey B. West who said something fishy about power laws last time!

Ctrl+F “power law”…

“Many diverse properties of cities from patent production and personal income to electrical cable length are shown to be power law functions of population size with scaling exponents that fall into distinct universality classes.”

Growth, innovation, scaling, and the pace of life in cities, Luís M. A. Bettencourt, José Lobo, Dirk Helbing, Christian Kühnert, and Geoffrey B. West

(Actually, I can well believe that some of the things they looked at do follow power laws. I certainly don’t think that they *all* do.)

*Bread & Kisses* is a short film by Katherine Fitzgerald about a mathematician who discovers love – I know, I know, you’ve heard this one before – but it also contains a mathematician who moves to the Alps to get more skiing in, so it’s the most realistic film about mathematicians ever. It also features the emotion of love in a star turn as an epsilon term.

Although it contains the line, “you forgot the most important ingredient: love”, so don’t get your hopes too high.

Mr. Maths is struggling with a proof. He signs up to a baking class because an Attractive Woman tells him to. In the process, he learns to loosen up and not overthink or something.

Or, in the words of the producers:

A lonely mathematician, struggling to solve an equation finds a solution in an unexpected way – by taking baking classes.

The two women who teach him awaken his senses, rekindle his zest for life and show him the value of balancing his head with his heart.

Because maths is about balancing equations, *amirite?!?!*

It came out in 2010 but just appeared on my radar recently. Maybe it’s only just been uploaded to Vimeo. Thanks to that, you can watch it here. It’s pleasant enough.

*via Luis Guzman on Google+*

To celebrate Christopher Zeeman’s 90th birthday and their own 150th, the London Mathematical Society have opened an online archive of Sir Christopher’s work.

That’s all they’ve done – the Zeeman Archive, as far as I can see, is a simple list of every document they’ve got, linking to PDF scans. It’s searchable, by title, medium, and date. I’d like a bit more presentation and information to put stuff in context.

Disconcertingly, a few of the papers have the scary JSTOR licence page at the front. I assume the LMS has cleared their redistribution with JSTOR, but you’d think they’d arrange at least a customised cover page.

Anyway, making stuff available for free and easy to access is always a good thing — at least they haven’t gone for one of those awful museum archive systems that puts all sorts of barriers between you and the content.

I’m not particularly familiar with Zeeman’s work so I just went for a browse. There’s a fun note called “Unknotting spheres in five dimensions”, and the famous Royal Institution Christmas Lectures are all there (well, there’s a link to the corresponding pages on the Ri Channel). The photos and letters categories are currently empty; maybe someone is working away at gathering them.

**Have a dig around: **The Zeeman Archive at the LMS

The sight of bumblebees roaming around British gardens, foraging for nectar, is common and comforting. The movement of these fuzzy bees between flowers and plants can often seem deliberate yet erratic. Charles Darwin was intrigued by “humble-bee” routines^{1}, and observed them with the assistance of his six children, but always regretted not attaching strands of cotton wool to the bees so he could follow them more easily^{2}.

Within the last decade there has been renewed interest from a number of collaborating researchers into studying bumblebees’ movement between flowers and their foraging techniques. The prevailing journalistic spin on this research seems to be ‘Bees solve the Travelling Salesman Problem – a problem that mathematicians and computers cannot solve’. This is unfortunate, not least because it is gleefully misleading, confusing various meanings of ‘solve’, but also it obscures a lot of the fascinating underlying scientific investigations.

Imagine a purely mathematical bee: she sets off from her nest with a list of $N$ flowers she must visit before returning home. She wishes to minimise the total length of her flight path between the flowers, so will always fly in straight lines, and never visit the same flower twice in an outing.

Once she has found the flower order that gives the path of minimal distance, she has found *the solution to this particular instance* of the Travelling Salesman Problem, or TSP (in two-dimensional Euclidean space, which is the only version of the TSP that I’ll mention outside the footnotes).

However, every such mathematical bumblebee must solve a different problem: each has a different list of flowers, with differing numbers of flowers in different locations.

If we were to program robotic bees to pollinate specific flowers efficiently, it would be nice if they could solve the Travelling Salesman Problem in general. That would mean creating an algorithm where, given *any* list of flowers, the robot always finds the shortest path. It’s conceptually easy to create such a general algorithm: just get the robot to visit the flowers in every possible order, keeping track of the shortest circuit it has found so far. The issue is that this is horribly inefficient, taking $N!$ flights to find the TSP solution for $N$ flowers. Mathematicians don’t know whether there is an algorithm that efficiently solves the TSP in every case.^{3}

For most purposes, in practice, you don’t need to know how to find the minimal solution efficiently in every case: it’s enough to have strategies or *heuristics* that help you find *approximate solutions *that are short and may be close to minimal — for instance, if you looked at each flower in turn and took the minimal distance to any other flower, and summed these lengths up, that would give you a lower bound for the distance of all circuits. So by finding a circuit with a length close to this lower bound, you would know you were even closer to the actual solution. This would be the case with our mathematical bumblebee visiting a list of flowers: the reward in an evolutionary sense is for finding a decent approximate solution without expending too much effort searching. Reducing the distance travelled by that final millimetre to the absolute minimum, while rewarding for mathematicians, doesn’t make much difference to the bees.

So far we’ve been talking theoretically. Real bees aren’t equipped with lists of flowers they must visit. There’s an extremely wide choice of suitable flowers and bees aren’t restricted in which they can choose. Not all flowers are equal, having all sorts of different properties, such as their colour, scent, and the varying amounts of nectar they produce, as well as changing over time. Bees don’t have perfect knowledge of all flowers or the distances between them, and are quite limited in brain power and memory. Also, bumblebees don’t necessarily fly in straight lines.

Having said all this, we could still reasonably expect that whatever heuristics bees develop or have hard-wired into their brains^{4} to solve their own routing problems might also do quite well at solving the Travelling Salesman Problem. Even if the TSP doesn’t model the bees’ problems in an ideal way, at least the TSP describes a precise problem, and is used in experiments with other animals. Therefore, using the TSP allows comparison of results between species with wildly different environments and behaviours.

So how might you go about designing an experiment to convince bees to follow the rules and solve a Travelling Salesman Problem? For starters, you can restrict the number of flowers available to the bees by performing the experiment in a greenhouse, or in a field otherwise lacking in flowers.

Another strategy is using artificial flowers, to control how much nectar (or rather sugar syrup) each provides. It is good practice here to get an initial baseline measurement of how much a bumblebee’s crop can hold, and then set each of the $N$ artificial flowers to supply $1/N$ of the total nectar the bumblebee requires. This removes variation between flowers, and encourages the bumblebees to visit each flower at least once. As for the problem of bumblebees visiting flowers more than once in a trip, in practice it turns out that usually the bee returns to the most recently visited flower, and these revisits can be ignored.

Finally, we shouldn’t expect the bumblebees to perform well while they are getting used to the layout of the flowers, but we should allow them to perform many foraging flights over the same arrangement to give them a chance to learn.

Bumblebees will often settle upon particular routes between a chosen set of food sources. This behaviour, known as trapline foraging, is shared by other creatures such as hummingbirds, bats, wagtails and capuchin monkeys^{5}. If the traplining bee has reliable and renewing sources of nectar, why bother searching for new ones? Visiting in the same order during each foraging flight reduces variation in the time between successive visits, and so the amount of sugary reward is more predictable. Regularly revisiting flowers reduces the benefits received by any competitors interloping onto the bee’s patch. However, the inevitable trade-off is that visiting the same flower too frequently also caps the rewards enjoyed by the foraging bee herself.

A couple of studies approximately following the above design were set up to test whether bees might follow a *nearest-neighbour* strategy. Following the nearest-neighbour strategy would mean the bees always head next to the closest flower they haven’t yet visited.

The nearest-neighbour strategy is an instance of the *greedy algorithm*, which means always taking the option at each stage that gives the greatest immediate reward. Specifically, the greedy algorithm involves no planning ahead, and depending on the situation, might not necessarily result in the greatest overall reward. Sometimes it goes completely wrong. In the Euclidean Travelling Salesman Problem, the greedy algorithm typically gets reasonably close to the optimal solution.

As an example of the greedy algorithm, when a UK cashier is handing you back change (1p, 2p, 5p, 10p, 20p, 50p, £1, £2), they can simply choose the largest possible coin not greater than the amount of change needed, subtract it off the total, and do the same with the next. This simple greedy strategy will always result in the smallest possible number of coins handed back.

That the greedy strategy works in this case depends on the values of the coins. The same wouldn’t hold if the cashier were handing back 6p using a selection of historic currency, namely pennies, threepennies (3p) and groats (4p). The greedy strategy here would lead to handing back three coins (4p, 1p, 1p), instead of the optimal two (3p, 3p).

A prime example of where the greedy algorithm fails is the board game Reversi (also marketed as Othello). It’s tempting to be greedy and each turn choose a move that flips over the greatest number of your opponent’s pieces to your own colour. This turns out to be a terrible strategy, especially near the start of the game. You can pit yourself against a silly computer opponent (“Simple Bot”), which uses a version of this strategy.

A greedy approach will fail in many strategy boardgames because human opponents can exploit such a simplistic and predictable strategy, and make short-term sacrifices to improve their long-term position. Equally, if the greedy algorithm is ever optimal, the game won’t be strategic or much fun, and can easily turn into a boring mechanistic chore.

Because the greedy nearest-neighbour strategy is simple and works quite well at routing problems, it seems a reasonable hypothesis that bees might use it. However, while flowers aren’t setting out to trick navigating bees, researchers are.

The diagram below depicts two arrangements of artificial flowers (black circles) and two possible routes the bees could decide to take starting and ending at their nest box (white circle). The top row shows the shortest circuit around the flowers, the lower row shows the greedy route a theoretical bumblebee subscribing to the nearest-neighbour strategy would take in each case. In Arrangement 1, the nearest-neighbour route is the shortest. In Arrangement 2, the distance between the two rows of flowers has been decreased so the optimal and greedy routes no longer coincide.

These arrangements were both used in the greenhouse laboratory tests described in the paper “Trapline foraging by bumble bees: IV. Optimization of route geometry in the absence of competition”^{6}. The results weren’t conclusive. In Arrangement 1, the bumblebees mostly followed the optimal greedy route. But in Arrangement 2, while the bees didn’t follow the optimal route, they ended up taking a ‘noisy’ variety of suboptimal routes (and not just the greedy one). While this doesn’t point to any particular strategy on the part of the bumblebees, it does suggest that their approach worked well in the arrangement where being greedy paid off, but was more confused when this wasn’t the case.

A later paper, fronted by insect cognition researcher and bee TSP veteran Mathieu Lihoreau, dispels the notion that bumblebees are purely greedy beasts. A flower arrangement was chosen that more severely punished any bees that acted out of strategic greed. Most (6 out of 8) of the bees being tested used the shortest route as their main route (20% of the time), while the other two bees used it as their second most common route. None of the bees used the purely greedy route more often than by chance.

One thing that was clear from this set of experiments is that the bumblebees learnt to take shorter routes over time, each continuously reducing the total length of its route over the 80 trips. The picture below shows the paths taken by one of the six bees that took the shortest route more frequently than any other route.

Lihoreau and two other researchers attempted to mimic the behaviour of bees with computer simulations. Their best attempt led to an algorithm that essentially suggested the bees experiment with different paths between flowers, but prefer hops that had previously lowered the overall distance travelled. The initial probabilities assigned to moving between two flowers in their simulation, the *transition probabilities*, were inversely proportional to the squared distance between the flowers. This means that the greedy route would initially be the most probable before any learning took place. Each time the bee happens upon a shorter route, all the transition probabilities that went into making up this route are multiplied by a number known as a *learning factor* (1.1 worked well for them) and then normalised so the probabilities sum to 1. Thus the shortest transitions between flowers start off as the most highly weighted, but when the bee finds a shorter route, the hops between flowers on this route are more likely to be used in future.

Another property of this heuristic is that it is *scale invariant*: if all the distances between the flowers were multiplied by a constant factor, the model bee would act no differently. Since real bees are going to expend more energy taking suboptimal routes over larger distances, we would expect them to be incentivised to learn more quickly. To account for this, the learning factor can be increased. Increasing the learning factor won’t always lead to improved performance; it simply lowers the amount of risk-taking. If the learning factor is increased too much, the heuristic degenerates into an ‘order of discovery’ rule of thumb, an easy-to-falsify hypothesis that the bees just visit flowers in the order that they discovered them.

Using an increased learning factor of 2, the model did still fit larger scale data collected by Lihoreau and his collaborators when they took the experiments out of the greenhouse laboratory and into a field in Hertfordshire. Attaching tiny radar transmitters to the bumblebees’ backs, the researchers were able to more accurately map out the bumblebee paths. Using radar data, one can depict precise paths between the flowers, dropping the assumptions about direct flights. For instance, initial flights were extremely wiggly, with lengths of about 2000m, while still missing and repeating some of the artificial flowers. Before 30 trips were over though, one bee had optimised its route to 365m, just over the minimum of 312m, visiting flowers in the ideal order and in almost straight lines.

This model, unlike the real bees, will eventually settle permanently on one particular route. Though bumblebees are generally creatures of habit and can be easily tricked into taking suboptimal routes^{7}, they do occasionally try out a radically different route, and the converging computer model fails to capture this behaviour. A bee’s thirst for exploration makes sense when faced with a changing environment.

Recreating Lihoreau et al’s algorithm, I’ve generated one possible set of paths that one such simulated bee would take, learning with the same set of flowers as the real bee over the course of 80 foraging trips. At first glance, other than the fact that the simulated bee always visits all the flowers, the paths look quite similar. Both the real and virtual bee visit the optimal route early on, and by the end of the experiment are predominately using it.^{8}

One difference I see is that the simulated bee seems predisposed to use the greedy path more frequently in this flower arrangement than the actual bee. The greedy path only differs here by one ‘choice’ from the clockwise optimal route. You can also observe in these cases that, unlike the computer-generated bee, the real bumblebee is still experimenting with different routes in the final row, even though it’s been using the optimal route more by that point.

It’s easy to find criticisms of any specific algorithm modelling the way bees choose paths between flowers. After all, bee brains are very dissimilar to our mechanical, designed computers and the programs that run on them. But what Lihoreau et al. have shown is that a simple algorithm will adequately capture many aspects of a bumblebee’s behaviour, without attributing any particular sensory or computational superpowers to them. In the end, you could conclude ‘OK, it’s probably not *this* algorithm that bees are using, but something that feels a bit like it’.

**Apiological Part 1: **Honeycomb geometry

**Apiological Part 2: **Estimating nest volumes

- Bumblebees were generally known as “humble-bees” until the modern term really caught on in the 1890s.
- Freeman,
*Charles Darwin on the Routes of Male Humblebees* - There probably isn’t an efficient general algorithm: the 2D Euclidean TSP and wider classes of travelling salesman problems with other metrics, or non-complete graphs are
*optimisation problems,*and they are NP-hard. Informally, it’s hard even to check whether a suggested solution to a particular problem is minimal. These should not be confused with*decision problem*variants, also confusingly known as travelling salesman problems: find out whether a particular finite graph has a Hamiltonian circuit; does a graph have a Hamiltonian circuit shorter than some given distance $d$. These decision problems are NP-complete, as it’s easy to check whether a proposed solution meets the requirements (ie. in polynomial time). For instance, in the second decision problem you need only to check whether the length of the circuit suggested as a solution is greater or less than the given distance $d$. - Note the mathematical non-exclusive ‘or’ – bees might be hard-wired to later develop certain heuristic behaviours in some environments.
- Ohashi, Thomson:
*Trapline foraging by pollinators: its ontogeny, economics and possible consequences for plants* - Trapline foraging by bumble bees: IV also tested whether bees might be biased to moving in straight lines – they weren’t.
- Another experiment showed that when the last flower in a learnt optimal circuit was rigged to give more nectar, the bumblebees would sensibly prioritise this highly-rewarding flower, though at the expense of taking a longer route – even though they could have both achieved the same route length and visited the highly-rewarding flower first, by simply flying their original route backwards.
- Behold! Using my computer I’ve solved the Travelling Salesman Problem, a problem that even mathematicians with computers are unable to… oh wait, there might be a flaw in the newspaper logic used here.