I thought it might be interesting (to me, at least) to list the types of assessment I’ve been involved in marking in the 2015/16 academic year.

These are not all of my invention (i.e. some are things I made up in teaching I ran, others are pieces I delivered as part of some else’s design). In no particular order (numbers are approximate):

- 120 short individual tests (four tests times thirty students) — a series of short, unconnected questions;
- 16 multiple-choice tests;
- 32 group activities (four activities times eight groups) — students had to solve a slightly open-ended question as a group and I marked them on the written description of their solution and how well they had communicated and worked as a group during the task;
- 266 short individual courseworks — well, one was not particularly short, but they were all a series of short, unconnected questions;
- 30 in-depth individual courseworks — this had a series of connected and increasingly open-ended questions to investigate a topic;
- 6 group essays — students worked in groups to research history of maths topics and wrote their findings as a short (500 word) essay plus a brief (100 words) account of their estimation of the reliability of the sources they used; they did this formatively weekly for half a term before handing one in summatively;
- 25 individual history of maths essays — topic of student’s choice (with agreement);
- 15 group presentations accompanied by two-page handouts — this was to describe the findings of an open-ended group investigation;
- 25 group project plans and minutes of 75 group meetings — for the above investigation;
- 99 self- and peer- reflections on contribution to group work — for the same;
- 36 reflective personal statements discussing career plans, skills relevant to those and ethical issues;
- 10 individual presentations — interim reports on final year projects;
- 6 dissertations — final reports of year-long final year projects, each with a corresponding viva;
- 4 group presentations — to report on findings of a semester-long, open-ended group investigation;
- 16 group posters — to report on the above investigations;
- 1 group report — report of the same;
- one quarter of the questions on 200 group-marked exam scripts (two exams).

Once I was happy with the proof, I decided to record a video explaining how it works. Here it is:

*I probably made mistakes. If you spot one, please write a polite correction in the comments.*

Apparently those symbols winding their way around the garden are “plant growth algorithms”, whatever those are.

There’s also a golden-ratio-thingy water feature, of course.

You can thank Winton Capital, sponsors of all sorts of worthy maths projects, for this bit of mathsy art.

]]>**Theorem: **every 5-connected non-planar graph contains a subdivision of $K_5$.

The above statement, conjectured independently by Alexander Kelmans and Paul Seymour in the 70s, is very easy to say. And the video below, starring Dawei He, Yan Wang, and Xingxing Yu, makes it look very easy to prove:

It’s like they got Wes Anderson to film an academic PR video. In the normally uninspiring world of maths press releases, it’s quite refreshing. And the written press release is pretty snappy, too. Let’s not make this a *thing*, though.

However, as “one of those maths whizzes out there”, I wanted to know a bit more about the work than a two-minute video can impart, so I’ve looked up the working-out. There’s a pair of papers building up the proof: “The Kelmans-Seymour conjecture I: special separations”, and “The Kelmans-Seymour conjecture II: 2-vertices in $K_4^{-}$”. They’re decidedly *not* as aesthetically pleasing as the video: here’s an excerpt from paper 2:

Maybe a publisher will *add value* to the paper in the form of some line breaks.

Anyway, I thought I’d already been told this theorem as a fact, so congratulations to Dawei He, Yan Wang, and Xingxing Yu for finishing off such a lovely theorem. That’s assuming the proof works: so far, there are just the two preprints on the arXiv and a press release from Georgia Tech. I haven’t been able to find anything from other experts in the field to add credibility to the claim of a proof.

40-Year Math Mystery and Four Generations of Figuring – press release from Georgia Institute of Technology.

**Read the papers****: ** The Kelmans-Seymour conjecture I: special separations, and The Kelmans-Seymour conjecture II: 2-vertices in $K_4^{-}$. There isn’t much there for the tourist, though.

Finally, I can’t restrain myself from pointing out that paper 1 cites a paper by my maybe-relative, Hazel Perfect, “Applications of Menger’s graph theorem”. So cool!

]]>(For once I can use an exclamation mark next to a number without wise alecks making the canonical joke)

Maths and stats! On BBC Radio 1! Who’d’ve though it!

DJ Clara Amfo and the ubiquitous Hannah Fry have got a new series on the UK’s top pop station, looking at music from a mathematical perspective.

*Music by Numbers* (excuse me, *Music by Num83r5*), is currently being broadcast at 9pm each Tuesday, and there are a couple of episodes already on iPlayer Radio to catch up on. The first is about Coldplay (records sold: millions; distinct tunes composed: 1) and the second looks at a few numbers to do with Iggy Azalea’s career.

It’s mostly a very easy listen, more a biography hung off a list of numbers than any real maths, but that might be your cup of tea. And Dr Fry’s segments do go into a little bit of depth about subjects like how the top 40 chart is calculated.

I’ll warn you now that each episode is an hour long, with a lot of music breaks. If you’re like me, your tolerance for some of the featured artists might not be sufficient to get through a whole episode in one go.

**Listen**: *Music by Numbers* on BBC Radio 1.

Since 2010, I’ve been maintaining a list of “interesting esoterica” – papers, books, essays and poems that I find interesting entirely on their own merits. It’s mainly bits of esoteric maths – hence the name – but I’ve also included quite a few things just because they have amusing titles. The main idea is that when I’m talking to someone and want to show them a cool thing that I’ve half-remembered, I can look up the exact reference: I’ve shared the paper “Orange peels and Fresnel integrals” more times than I can count (probably the same as the number of times I’ve eaten an orange).

Back when I started the collection, I used the free “research manager” Mendeley to keep track of things – it offered an easy way to press a button while looking at a paper and instantly save all the right metadata about it to my personal library. Over the years, the experience of using Mendeley has either got worse or not improved as much as I’d like it to: I use my desktop PC less than I used to, and the desktop app hasn’t improved much anyway; the web interface to my library has steadily lost features as they rewrote it; and because I set the Interesting Esoterica collection up as a group, Mendeley won’t let me download copies of PDFs that I uploaded to it. There’s always been the little niggle that I’m giving data to Mendeley, a private company (and now a subsidiary of Elsevier), who can decide what I’m allowed to do with it.

And what I want to do with my Interesting Esoterica collection isn’t what Mendeley is normally used for – I’m not writing research papers which need to have citations formatted in the correct style; I’m not working with a group of co-authors; and some of the stuff I’m collecting isn’t classic “research material” at all. What I *do* want is to present the things I’ve found nicely – the collection is an aesthetic project as much as anything.

While scraping the dates, I noticed that the Mendeley web interface uses American-style dates. How unenlightened!

So, I decided to set up my own site to manage and present my collection. Fortunately, it wasn’t too much effort to get most of my data out of Mendeley – after reinstalling the desktop app, I could generate a .bib file of my collection, which contained almost all of the data I wanted to keep, apart from the dates I added each item to the collection, and my library of saved PDFs for papers that aren’t available for free on the web. To get the “added on” dates, I opened up the Mendeley web interface and wrote some javascript to scrape the dates off the page. There might be a way to get this information by using their API, but this was much easier.

I decided that keeping all of my information in a .bib file would be the best way to keep it usable, no matter what happens in the future. The BibTeX format does what I want anyway – record references to things I’ve seen – and if I’d made up a database scheme, I’d have had to think about how to generate a .bib export anyway.

The hosting service that I use for my personal sites, as well as The Aperiodical, only supports yucky PHP scripting, not lovely Python, so I set about hacking together something which would parse my .bib file and display its contents in a web page without being able to rely on any of the good existing Python libraries to process BibTeX. There are a few PHP libraries which claim to parse BibTeX, including phpBibLib, which I’ve used before for a work thing. On running it over the file I exported from Mendeley, however, I discovered that it doesn’t *really* parse BibTeX, it just does some global string replacements, and some of the fruitier accented character commands in my file broke it. I took a look at the source code to see if I could fix it, but decided to start again from scratch: to get phpBibLib to parse properly would involve effectively rewriting it anyway, and it isn’t released under a licence that would let me share my changes easily.

Fairly shortly I had a script which loaded my .bib file and rendered a summary in HTML. My first attempt at writing a formal parser using a PEG library was far too slow to be usable, but once I’d written that I was able to rewrite a more low-level parser which is fast enough to parse the entire file each time a page is loaded. If you’re interested in using my parser, it’s on my github repository.

Next, I added forms to edit existing entries and add new ones. One of the great attractions of Mendeley was that I could use a bookmarklet to add the page I’m looking at to my library, so that was the first feature to go in. Then it turned out that the Mendeley data wasn’t that great – papers from the arXiv didn’t always have the right metadata and links to PDF versions attached, and a few other entries must have just been PDFs that I put in the desktop app, so I had to track down relevant URLs based on their titles. That took a couple of hours.

Finally, I could work on making the site look nice. I made the index page list all the entries in a patchwork design, with each item painted using a different colour drawn from a desaturated palette. On the detail page for an individual entry, I made sure the title, authors and abstract were prominent, and when the entry had a reference to a PDF file I embedded it on the right hand side. To be completely open and make sharing even easier, I show the BibTeX for each entry at the bottom of the page, and there’s a link to download the whole .bib file at the bottom of the index page. In an inspired last touch, I added an “I’m feeling scholarly” button next to the search form, to take you to a random entry. That’s my favourite thing about the collection: whenever you pick a random item from it, you’ll find something really interesting and entertaining. I’ve had a lot of fun rediscovering things that I’d added a few years ago and almost completely forgotten.

So, that’s the story of how I liberated my data from a benevolent cloud service, and made a site to present it just the way I want. You can have a look at my Interesting Esoterica collection at its new home, **read.somethingorotherwhatever.com**.

*If you’d like to do something similar with your .bib file, I’ve put all my code on GitHub at github.com/christianp/bib-site. I’ve set it up so you can easily change how the site looks, without meddling with PHP code. All you need is a web host that can run PHP – no database or weird CGI stuff required.*

Manchester Science Festival’s mass-participation maths/gardening project, Turing’s Sunflowers, ran in 2012 and invited members of the public to grow their own sunflowers, and then photograph or bring in the seed heads so a group of mathematicians could study them. The aim was to determine whether Fibonacci numbers occur in the seed spirals – this has previously been observed, but no large-scale study like this has ever been undertaken. This carries on the work Alan Turing did before he died.

The results of the research are now published – a paper has been published in the Royal Society’s Open Science journal, and the findings indicate that while Fibonacci numbers do often occur, other types of numbers also crop up, including Lucas numbers and other similarly defined number sequences.

Manchester Science Festival – Turing’s Sunflowers project

Novel Fibonacci and non-Fibonacci structure in the sunflower: results of a citizen science experiment, Royal Society Open Science

Earlier this month the Royal Society announced their new Fellows, which include several mathematicians.

- Algebraicist and maths mascot Marcus Du Sautoy
- Topologist/dynamicist Caroline Series
- Oxford University geometer, topologist and group theorist Martin Bridson
- Gerd Faltings from the Max Planck Institute
- Economist Adair Turner
- Statistician Christl Donnelly
- Applied Mathematician Lakshminarayanan Mahadevan

You can also view the full directory of Royal Society Fellows, and the sublist of Fellows elected in 2016.

Katherine Johnson is a mathematician who worked for NASA at the height of the space race. Her work included computing (by hand!) trajectories for early rocket flights and influences spaceflight even today, for which she was given the Presidential Medal of Freedom in 2015 by Barack Obama.

The new Katherine G. Johnson Computational Research Facility was formally dedicated to the venerated mathematician earlier this month, and houses advanced computational research and development in a 40,000-square-foot consolidated data centre.

It’s also been announced this year that *Hidden Figures*, an upcoming film about African-American women working at NASA will feature Taraji P. Henson as Johnson, as well as Janelle Monáe and Oscar winner Octavia Johnson. The film will be out in 2017.

NASA Dedicates Facility to Mathematician, Presidential Medal Winner, NASA Langley Research Centre.

Katherine Johnson, the girl who loved to count

She was a computer when computers wore skirts, NASA Langley Research Centre.

NASA on flickr

Hidden Figures, on IMDB

Janelle Monáe & Taraji P. Henson To Star In Film About Black Women In NASA, at Vibe

The L-Functions and Modular Forms Database is a huge reference for data on all sorts of number-theoretical objects. It’s been going for a little while, but the creators recently declared the site “ready to use”, and went on a little press blitz.

In short, the Riemann zeta function is a kind of L-function, so one of the reasons people look at L-functions is the chance that doing so will help them make progress on the Riemann hypothesis. None of us understand enough about the maths involved to explain the LMFDB in any more depth, but plenty of people who *do* understand have written blog posts about it, so I’ll just link to those.

Numberphile made this video talking about how L-functions relate to the Riemann hypothesis:

The site itself: L-Functions and Modular Forms Database

International team launches vast atlas of mathematical objects press release from MIT.

The L-functions and modular forms database project paper describing the project, by John Cremona.

L-functions database! at E. Kowalski’s blog.

LMFDB! at Jordan Ellenberg’s blog.

L-functions and modular forms database at Timothy Gowers’s blog.

]]>

“I’m proud that I’ve lived to see… so many of the things that I’ve worked on being so widely adopted that no one even thinks about where they came from.”^{1} Solomon Golomb (1932-2016)

Solomon Golomb, who died on Sunday May 1st, was a man who revelled in the key objects in a recreational mathematician’s toolbox: number sequences, shapes and words (in many languages). He also carved out a distinguished career by, broadly speaking, transferring his detailed knowledge of the mathematics behind integer sequences to engineering problems in the nascent field of digital communications, and his discoveries are very much still in use today.

His mathematical interests started with prime numbers and number theory, completing his thesis “Problems in the Distribution of the Prime Numbers” in 1957. Shortly afterwards,

while working on matters of pure mathematics that supposedly had no practical application, specifically number theory and advanced algebra, he became interested in communications and cryptography. He began to think about how a curious mathematical phenomenon called a nonlinear shift register, or pseudorandom sequence, could be applied in those fields.

^{2}

Golomb worked

as a Senior Research Mathematician at Jet Propulsion Laboratory, later becoming Research Group Supervisor and then Assistant Chief of the Telecommunications Research Section, where he played a key role in formulating the design of deep-space communications for the subsequent lunar and planetary explorations.

^{3}

In 1967, Sol Golomb wrote the book “Shift Register Sequences” about the predictable but seemingly random *non-linear shift register sequences*, which are

used in radar, space communications, cryptography and now cell phone communications. This book has long been a standard reading requirement for new recruits in many organizations, including the National Security Agency and a variety of companies that design anti-jam military communication systems.^{4}

To see the breadth of Golomb’s interests, you only have to look at his publications list. His impact on recreational mathematics becomes obvious if you browse through a collection of Martin Gardner’s Mathematical Games columns – he has inspired at least four of these in full, with numerous smaller contributions. These include:

**Polyominoes**

Solomon Golomb didn’t invent polyominoes^{5}, but did coin the terms*polyomino*and*pentomino*(even holding the trademark on Pentominoes for some years) and created a whole subject area around them^{6}, writing the defining 1965 book “Polyominoes” which was well-received and reached a broad audience: the 1975 Russian translation famously inspired Tetris. I recommend playing Golomb’s own game: players take turns placing one of the 12 pentominoes on a chessboard – the first player unable to make a move, loses.**Rep-tiles**

Solomon gave a quarter-hour overview of rep-tiles at the 2014 Gathering for Gardner, which you can watch here.**Cheskers**A variation of chess played on only one colour of squares of the chessboard, as in draughts/checkers

**.****Golomb rulers**

Again, studied for their applications and popularised, but not invented by Solomon, Golomb rulers were so named by Martin Gardner. Golomb also named Costas arrays, and coined a catchier name for a generalisation of Golomb rulers:*graceful labellings*(of graphs). Knowing Golomb’s passion for wordplay, many assumed that his co-author on several Golomb ruler papers, G. S. Bloom^{7}, was merely an invented anagram of ‘S. Golomb’, when it was actually his student Gary’s name (as Sol fondly recalled in his 2014 Word-ways column “Anagramming Co-authors“).

Leafing through the Martin Gardner collection “Fractal Music, Hypercards and More”, for instance, reveals many more minor Golomb contributions in articles on “Tangent Circles” (with a couple of coin puzzles from Golomb’s article “Wreaths of Tangent Circles”), “Square Packing and Tiling” (can you tile the infinite plane with squares of side lengths 1, 2, 3, …?), and “Mathematical Chess Problems” (can you place *n* ‘superqueens’, which combine the movements of queens and knights, on an *n* by *n* chessboard so that none threaten each other?).

Given Sol Golomb’s interest in integer sequences, it seems appropriate to mention a couple of his. The self-describing *Golomb’s sequence *made it onto sequence connoisseur and OEIS founder Neil Sloane’s list of favourite sequences:

1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8,… (A001462).

Solomon also investigated Hofstadter and Conway’s *strange recursions,* exploring simpler examples, and giving alternative starting conditions to one (the Fibonacci-ish $a_n := a_{n-a_{n-1}} + a_{n-a_{n-2}}$) to create a “quasi-periodic sequence of quasi-period 3”:

3, 2, 1, 3, 5, 4, 3, 8, 7, 3, 11, 10, 3, 12, … (A244477).

Beyond mathematics, Sol was interested in history, words and language (and could speak seven languages). His articles for the Word Ways recreational lingusitics journal show off his playfulness and knowledge on a range of topics: “A Letter to Martin” gives some obscure observations after a reading of Gardner’s “Annotated Alice”; alike-sounding ‘false-friends’ are explored in “Hebrew or Japanese?“; and “Extraterrestial Linguistics” draws together Golomb’s interests in language, space communications and mathematics.

Of course, nothing can give you a better sense of the man than listening to Sol himself talk about his life in an hour-long interview for the University of Southern California’s living history project, or watching a more mathematical and technical lecture he gave on his life’s work.

- Solomon W. Golomb – 2016 Laureate of the Franklin Institute in Electrical Engineering
- Benjamin Franklin Medal write-up (2016)
- Solomon Golomb’s University of Southern California faculty page (archived)
- Introduction to Solomon W. Golomb (2002)
- In “Polyominoes
*“*, Golomb attributes the observation that “there are twelve distinctive patterns (the pentominoes) that can be formed by five connected stones on a Go board… to an ancient master of that game”. Pentaminoes also appeared, for instance, in Dudeney’s “The Broken Chessboard” puzzle, published in the 1907 “Canterbury Puzzles“. - Golomb’s interest in polyominoes started as a student by generalising the mutilated chessboard problem.
- eg.
*Applications of numbered undirected graphs*, G.S. Bloom and S.W. Golomb

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.

]]>It’s Eurovision time again! A chance for everyone to enjoy musical performances that are either good or so bad they’re good, ridiculous staging, and hilarious costumes, all sprinkled with a gently sarcastic Irish voiceover (if you’re lucky enough to be watching in the UK).

BUT WHAT’S THIS? They’ve changed the voting system? Don’t worry – some mathematicians are here to straighten it out for you.

A hangover from an old attempt at a unified European TV channel, the **Eurovision Song Contest** takes place each year in the country that won it the previous year. That country, along with five other countries who put a lot of cash in (France, Germany, Spain, Italy and the United Kingdom) get automatic entry to the contest, and semi-finals are held in the week running up to decide which countries will make up the rest of the 26 entrants, whittled down from 40.

Under the old Eurovision voting system, each country, including eliminated semi-finalists, held a telephone vote on the night of the show, with the option to vote for any of the 25 or 26 other countries (no, you can’t vote for yourself). The results of that phone poll, alongside the votes of a jury of musical experts, were condensed down to a set of points given by each country – 12, 10 and 8 down to 1, ranking their top 10 countries in order. The points awarded by each country are announced by one if its minor celebrities, one country at a time over a patience-stretching hour-and-a-half or so, slowly revealing the final ranking.

Previously, we’ve written about the way the phone and jury information is combined to give the final points total, which is somewhat mathematically arbitrary, and never clearly explained or even really acknowledged.

Clearly, the Eurovision producers are avid Aperiodical readers, as they’ve rejigged this part of the system for the 2016 contest.

Under the new scoring system, jury and televoting results are kept separate, and each create their own ranking and set of 1, 2, 3, 4, 5, 6, 7, 8, 10 and 12 points; so twice as many points are awarded overall. The soporific 90-minute results drip-feed will only cover the results from the juries. After that, the results of the popular vote will just all be added on at the end in one go (the word ‘thwumpf’ has been used). The idea is that this keeps the results announcement more interesting: in past years it’s become mathematically impossible for any but the currently-leading country to win well before the end of the show. Since fully half the points are kept back until the end now, in theory any country could take 1st place at the last minute – if the callers at home are sufficiently contrarian relative to the expert juries.

This new system raises a couple of important questions. Is it likely to produce different results to the old system? And that big exciting announcement of the last half of the points – how much of a shake-up is it likely to create in the table?

Luckily, the kind bods at Eurovision provide the full jury and televote results for the last couple of years on their website, so we can simulate what would have happened if the new system had been in place. It’s time for SOME MATHS.

Here’s a table showing the top part of last year’s results:

Country | Score | Position |
---|---|---|

Sweden | 365 | 1 |

Russia | 303 | 2 |

Italy | 292 | 3 |

Belgium | 217 | 4 |

Australia | 196 | 5 |

Latvia | 186 | 6 |

Estonia | 106 | 7 |

Norway | 102 | 8 |

Israel | 97 | 9 |

Serbia | 53 | 10 |

Under the new system, each score would be split into a jury score and a televoting score, and the countries would be ranked in order firstly using the jury scores, then again after adding in the televoting results. Here’s how that would look:

For anyone wondering why Australia appears in this table, they were allowed to enter as a one-off in 2015 as part of the 60th anniversary celebrations of Eurovision. This was a total one-off, and won’t happen again. Except in 2016, when it’s happening again. But that’s probably it.

Country | Jury Score | Position based on jury score only | Televoting score | Total score from jury and televoting | Final position | Position under old system |
---|---|---|---|---|---|---|

Sweden | 353 | 1 | 272 | 625 | 1 | 1 |

Italy | 171 | 6 | 356 | 527 | 2 | 3 |

Russia | 234 | 3 | 286 | 520 | 3 | 2 |

Belgium | 186 | 5 | 190 | 376 | 4 | 4 |

Australia | 224 | 4 | 124 | 348 | 5 | 5 |

Latvia | 249 | 2 | 88 | 337 | 6 | 6 |

Norway | 163 | 7 | 37 | 200 | 7 | 8 |

Estonia | 53 | 11 | 144 | 197 | 8 | 7 |

Israel | 77 | 8 | 102 | 179 | 9 | 9 |

Georgia | 62 | 10 | 51 | 113 | 10 | 11 |

Serbia | 12 | 23 | 86 | 98 | 11 | 10 |

Armenia | 15 | 22 | 77 | 92 | 12 | 16 |

Romania | 21 | 21 | 69 | 90 | 13 | 15 |

Azerbaijan | 40 | 13 | 48 | 88 | 14 | 12 |

Montenegro | 44 | 12 | 34 | 78 | 15 | 13 |

Lithuania | 31 | 16 | 44 | 75 | 16 | 17 |

Cyprus | 63 | 9 | 8 | 71 | 17 | 21 |

Slovenia | 36 | 15 | 27 | 63 | 18 | 14 |

Greece | 29 | 17 | 24 | 53 | 19 | 18 |

Poland | 2 | 26 | 47 | 49 | 20 | 22 |

Hungary | 29 | 17 | 17 | 46 | 21 | 19 |

Austria | 40 | 13 | 0 | 40 | 22 | 25 |

Spain | 6 | 25 | 26 | 32 | 23 | 20 |

Germany | 24 | 19 | 5 | 29 | 24 | 25 |

France | 24 | 19 | 3 | 27 | 25 | 24 |

United Kingdom | 12 | 23 | 4 | 16 | 26 | 23 |

The most obvious fact here is that there’s not much difference between the rankings under the new system: we have the same winner, the same top 3 and top 5: in fact nobody moves by more than four positions. There would have been little change in 2014 either: the top 5 is place-for-place identical under the two systems.

So, we can rest easy that this switch probably won’t affect the overall results too much. But surely the big phone-vote reveal at the end is going to be total ranking chaos, rendering the entire jury reveal a pointless exercise?

In fact the data suggests not. There’s a lot of movement outside the top of the table – Serbia shoot up from 23rd to 11th and Cyprus crash from 9th to 17th – but the winner is unchanged. In 2014 the top 3 would all remain static, with Poland rising from 23rd to 6th the biggest change. In general we might suppose that ‘novelty’ acts are likely to shift up the table after failing to amuse the po-faced juries, but going down a treat with the more accepting/drunk public.

If you’d like to perform your own statistical analysis of the results of Eurovisions past, the data is all available on their website, and Excel spreadsheets can be downloaded for 2014 and 2015. Or, just watch it and cheer when the people are silly.

]]>