1. All four-digit palindromic numbers are divisible by 11.

This is quite easy and nice to prove. Start by writing the palindrome as a sum expressed in terms of the two different digits.

2. Repeat a three-digit number twice, to form a six-digit number. The result will be exactly divisibly by 7, 11 and 13, and dividing by all three will give your original three-digit number.

I found this in a Martin Gardner book. Can you see why it works?

3. Choose a digit from 1-9, and repeat it three times to give a three-digit number. If you divide this number by the sum of its digits, the answer will always be 37.

This is also fun to figure out. It’s an especially nice trick to pull out at a 37th birthday party.

4. If you ask someone to choose ten random digits (not including 0) and multiply them all together, then to tell you all but one of the digits of the answer, you can predict the remaining digit.

Can you guess why this is possible? It relies on the fact that people asked to choose random digits will normally hit enough of the right digits to make this work. It doesn’t work every time, but if you increase the number of digits you ask for, it makes it more likely to work. Hint:¹

I’ll post the proofs to all of these next week. Please refrain from posting your own in the comments, so people can try to work them out for themselves.

1. The sum of the digits of the number will usually be a multiple of 9.

The article discusses people’s misconceptions about the future utility of what they were learning, as well as the divide between using ‘any math’ and ‘advanced math’, which includes calculus, algebra and statistics. The number of Americans who admitted to using this type of maths appears to drop off once you get to anything more complicated than fractions, and also presented is an analysis of this divide by job type.

A very well-written and thoughtful response to this has already been posted at mathematics professor Bret Benesh’s blog, which gives four reasons why the article annoyed him (and probably several other people too).

If you enjoy getting outraged at people who are wrong on the internet, there was another article posted on Gizmodo the following day, entitled ‘Who Actually Uses Math At Work?‘, and using the same graph of data along with a scathing putdown of the entire subject of mathematics, including such gems as the opening statement,

Let’s admit it together. We all kind of suck at math. It’s okay! Numbers are evil.

The thrust of the Gizmodo piece is summed up by this quote:

But for someone who will never work with numbers in any sort of way passed addition and subtraction, why the hell are they taking advanced math classes every single year of high school?

Well, to start with, for someone who writes articles on the internet, maybe you should have skipped some maths classes and done more work on your spelling, but anyway, this argument was given short shrift by various people in the comments section. As James Grime pointed out on Twitter, the type of person who reads Gizmodo (a website of gadgets and technology news) is unlikely to sympathise with an anti-maths stance, and this level of maths-bashing is basically unnecessary.

A Twitter conversation between Matt Parker (@standupmaths) and several others went as follows:

Thus proving, at a stroke, both that sarcasm is the form of wit most likely to be deployed in the face of ridiculous statements, and that every conversation on the internet eventually reaches the point where someone posts a link to a webpage discussing NP-completeness and Turing-completeness of minesweeper.

Further reading:

Here’s How Little Math Americans Actually Use At Work, The Atlantic

Painters and Pure Mathematicians, Solvable By Radicals blog

Who Actually Uses Math at Work?, Gizmodo

As an example, Colin Beveridge sent us this classic from his doormat:

We’ll be awarding bonus points for inaccurate pie charts, exaggerated/meaningless bar sizes, the complete absence of axis label or scale, the use of ‘Can’t win here!’ and any other sneaky/incompetent features. Email your submissions to root@aperiodical.com, and watch out for a roundup post if we collect a sizeable pile.

Today is Euler’s $-306 \times e^{i \pi}$^th birthday, and Google have chosen to celebrate (despite ignoring several other prominent mathematical birthdays, including Erdős’s centenary – see the @MathsHistory twitter feed for a full list) by creating a Google doodle on their homepage.

For anyone who isn’t aware, this is when Google changes the image above the search box on the homepage at Google.com, so it still says ‘Google’ but using an appropriate image, which sometimes has built-in interactive elements. I thought it was worth pointing out some of the fantastic maths they’ve included in today’s doodle.

In celebration of the Swiss mathmo’s achievements, the Google doodle today includes:

• The formula for the Euler characteristic, $V – E + F = 2$, which relates the number of vertices, edges and faces of a spherical polyhedron. The doodle also features some polyhedra (icosahedron and tetrahedron). The formula also applies to a diagram of vertices, edges and faces drawn on a flat plane, or on a sphere – and a similar one can be used for on other surfaces, by subtracting 2 from the right hand side for each hole in the object (so on a donut, $V – E + F = 0$).
• The famous ‘Seven bridges of Königsberg‘ problem, drawn as both the layout of bridges and a graph. Euler pioneered graph theory; an Eulerian graph is one which can be fully traversed starting from any point (equivalently, one where each node has an even number of arcs coming into it). Since the ‘bridges of Königsberg’ graph has an odd number of arcs coming into every node, it can’t be traversed from any start point – although if a graph has all even nodes apart from two, it’s called semi-Eulerian, and can be traversed only if you start at one odd node and end at the other.
• Euler’s identity, $-1 = e^{i \pi}$. This is often described as one of the most beautiful equations in mathematics – if you rewrite it as $e^{i \pi} + 1 = 0$, you have a relation involving five amazing constants (the additive and multiplicative identities, $0$ and $1$, plus the exponential constant $e$, circle constant $\pi$ and the imaginary number $i$) as well as all the major mathematical operations – addition, multiplication (between $i$ and $\pi$) and exponentiation. This relationship expresses nicely the connection between trigonometry and the exponential function, and it’s boss as heck.
• There’s also a geometrical interpretation of Euler’s formula, which shows how as the angle changes, the real and imaginary parts of the points on a circle in the complex plane change according to trigonometric functions.
• At the centre of the Google doodle is a three-dimensional representation of a sphere, which you can drag around and it will rotate in all three axes. Within the sphere you can see three orthogonal sections, creating a cross in each plane. Euler introduced the idea of Euler angles, to describe the orientation of a rigid body using an angle of rotation around each of the three Cartesian axes. This way of describing angles is widely used, from Euler’s time right through to modern day technology such as accelerometers – these are used in mobile computing devices and controllers to measure the position and motion in space of the object, which can be used to control the device (think Wiimotes, and tilty marble games on your phone, and then think of Euler and be grateful.)

Props to Google for popularising Euler’s amazing work, and for making such a lovely doodle! Also, Google, if you’re looking to employ someone to research and create amazing mathematical doodles like this in future, I’m totally available. Although I imagine there are enough Google employees interested enough in maths that they don’t have a problem there.

The volume of a sphere inscribed in a cylinder is 2/3 the volume of the cylinder. ow.ly/j3HjS #math #geometry #Archimedes

— Geometry Fact (@GeometryFact) March 22, 2013

Fact fans and Follow Friday readers will be aware that there’s already an excellent selection of mathematical-fact-tweeting accounts, including @TopologyFact and @AlgebraFact. Since the only worthwhile type of mathematics is pure maths, another important one to mention is @GeometryFact, which didn’t exist at time of last writing and came into existence about two weeks ago. If collinear points in a plane make your conic section go all convex, this is the fact feed for you.

2. @CutTheKnotMath

My wife just noticed that March 2013 has 5 Fridays, 5 Saturdays, and 5 Sundays. Have you, or your wife?

— Alexander Bogomolny (@CutTheKnotMath) March 20, 2013

If you’re aware of Cut The Knot‘s fantastic website of mathematical puzzles and facts, you’ll be pleased to hear that its author Alexander Bogomolny has a Twitter account, which he uses to tweet regular interesting stuff and mathematical musings.

3. @CarnivalofMath

Carnival of Mathematics 96 bit.ly/XxMUby

— The Carnival of Math (@CarnivalOfMath) March 2, 2013

If you’d like to be sent a link roughly once a month to a nice blog post rounding up nice maths blog posts from the intervening month, which you may or may not have already seen, then you should be aware that we host the Carnival of Mathematics here at the Aperiodical, and it has a Twitter account which reminds you when each new edition is out. If you’ve written or seen a good blog post this month, please send it in! The deadline for this month is 1st April.

That’s all for this edition of Follow Friday! Keep your eyes peeled for more, at some point when it’s Friday.

Probably the greatest mathematician of the twentieth century, Paul Erdős … was so eccentric that he made Einstein look normal. He was 11 before he ever tied his shoes, 21 before he ever buttered toast, and died without ever boiling an egg. Erdős lived on the road, traveling from conference to conference, owning nothing but math notebooks and a suitcase or two. His life consisted of math, nothing else.

- Clifford Goldstein, in The Mules That Angels Ride (2005), p. 125

Born in Hungary in 1913, Paul Erdős was the one of most prolific mathematicians of all time: the honour of most pages published goes to Leonhard Euler, but Erdős has the most papers published in collaboration with others – around 1,525. He worked as a nomad, travelling between conferences and other mathematicians’ hospitality, and wherever he went he created fantastic collaborations with the world’s greatest mathematicians.

Another roof, another proof.

- Erdős’ motto, as quoted in A Tribute to Paul Erdős (1990) edited by Alan Baker, Béla Bollobás, A. Hajnal, Preface, p. ix

Erdős worked in many areas of pure mathematics - combinatorics, graph theory, number theory, set theory and probability theory to name a few. He collaborated with 511 different mathematicians, and was always travelling – his possessions fit in a suitcase, and most of his earnings and awards were generally donated to charity.

Erdős knows about more problems than anybody else, and he not only knows about various problems and conjectures, but he also knows the tastes of various mathematicians. So if I get a letter from him giving me three of his conjectures and two of his problems, then it’s sure that these are exactly the kind of conjectures and problems I’m interested in, and these are exactly the kind of questions I may be able to answer.

- Béla Bollobás, of Trinity College, University of Cambridge in N Is a Number: A Portrait of Paul Erdős (1993)

Ask anyone to name a theorem, and they’ll probably come up with one of the really famous ones, like Pythagoras’ theorem. This super-handy hypotenuse fact states that for a triangle with sides A, B and C, where the angle between A and B is a right angle, we have $C^2 = A^2 + B^2$. This leads us on to a nice bit of stamp-collecting – there are infinitely many triples of integers, A, B and C, which fit this equation, called Pythagorean Triples.

One well-known generalisation of this is to change the value $2$ to larger values, and go looking for triples satisfying $C^n = A^n + B^n$. But don’t – Andrew Wiles spent a good chunk of his life on proving that you can’t, for any value of $n>2$, find any such triples. The statement was originally made by Pierre De Fermat, and while Fermat famously didn’t write down a proof, it was the last of his mathematical statements to be gifted one – hence the name ‘Fermat’s Last Theorem’ – and proving it took over 350 years.

This series is of course about the unknown results in mathematics, the mysterious cases of questions we don’t yet have an answer for – so I’ll take you in another direction. If instead of considering these Pythagorean triples as the sides of right angled triangles, we think of them as the sides and diagonal of rectangles (yes, I know that’s the same thing, but bear with me) – we can extend this question into a nice puzzle. These rectangles made from Pythagorean triples will all have all four sides with integer length, and also their diagonal is an integer.

If I want to find not just a rectangle whose sides and diagonals are all integers, but instead a three-dimensional cuboid, with all the side lengths as integers, and additionally the diagonal of each face an integer too, it’s a more difficult question. If you’d like, go away and write down the equations for the lengths of all of these things, and see if you can find a set where all the answers are integers. I’ll hide one possible solution in the image on the right (click on it to see the answers).

Such a cuboid, with all sides and face diagonals having integer length, is known as an Euler Brick. Named after Leonhard Euler, there are infinitely many of these – obviously, since having found one you can multiply all the lengths by an integer and get a larger one. A primitive Euler brick is one for which all the edge lengths are coprime. You can also take an existing Euler brick with edges $(a,b,c)$, and create another one with edges $(bc, ac, ab)$ which will also be an Euler brick.

However, it still hasn’t quite got interesting enough. Some of you may be ahead of me here. If you additionally require the internal corner-to-opposite-corner diagonal of the cuboid, called the space diagonal, to have integer length (I’m going to pause and let you appreciate how good the name ‘space diagonal’ is, for anything, ever) – this restriction creates an object called a perfect cuboid. Again, knock yourself out working out the set of equations you’d need to satisfy. Any perfect cuboid will necessarily also be an Euler brick. But here’s the crazy thing – no known example of a perfect cuboid exists!

According to Wikipedia (as of November 2012), nobody has found any solutions to the equations defining a perfect cuboid. Computer programs have been run which show that if any such cuboid exists, its smallest side must be longer than $10^{10}$ and one of its edges must be longer than $3 \times 10^{12}$. There’s also a nice list on Wikipedia of properties that the edges of a primitive perfect cuboid must satisfy, which are implied by modular arithmetic.

So there you have it! A problem so simple you could explain it to a child, and yet mathematics is stumped. Maybe a perfect cuboid will be found by someone, with a big enough computer, in our lifetimes. Perhaps, with enough mathematical ingenuity, someone will prove that there’s no such thing. Or maybe, if we’re really lucky, someone will find a use for all this. But that’s just a pipe dream.

References:

Fermat’s Last Theorem, on Wikipedia

Perfect Cuboid, at Wolfram Mathworld

Bill Durango’s “Integer Brick” Problem

Euler Brick, on Wikipedia

If anyone still hasn’t sorted themselves out with a calendar for 2013 – come on people, it’s February! – there’s a nice example of one here. It’s a dodecahedron which, once assembled, you can presumably orient to display the correct month (or the incorrect month, if you’re an impish sort).

The best thing about it for fans of LaTeX (the majestic mathematical markup language of kings) is that this thing is written entirely in LaTeX, using the TikZ package to create the graphics.

Download: PDF and TeX files, as well as all necessary packages, are available from TeXample.net.

(If you can’t compile the calendar yourself and want an up-to-date version, click on the Open in writeLaTeX link).

Some mathematics, pictured here being hard to illustrate in news coverage

As the heady excitement of the dawn of a forty-eight-Mersenne-prime world dims to a subdued, albeit slightly less factorable, normality, I have taken the opportunity to see what we can learn about the British press’s attitude and ability when it comes to the reporting of big numbers ending in a 1.

Overseas readers may not be aware that the UK’s public service broadcaster, the BBC, is funded by a mandatory annual £145.50 tax on all television-owning households. Therefore, it would be disappointing if some of these funds were not channeled into reporting the discovery in at least five or six separately-produced broadcasts across the organisation’s various radio and television outlets.

I’ll start with BBC radio’s flagship news programme, Today. At 7.20 in the morning on Friday 8th, they ran an interview with the prime’s ‘discoverer’ Curtis Cooper, introduced and conducted by presenter Justin Webb. This was probably the strongest of the $2^2-1$ interviews Cooper conducted for the BBC; Webb asked the most straightforwardly sensible question: what does one actually ask the computer to do to find the primes? Cooper gave a good overview of the maths behind the GIMPS project. The interview benefitted enormously by departing from Today’s house style and not bringing in a second interviewee forced to somehow argue against the large number, in order to inject a pointless and counter-informative facade of balance into proceedings.

Cooper popped up again the previous evening (having finally obtained a prime number large enough to serve as a power source for his time machine) on the BBC World Service’s Newshour. Showing that nobody is immune to doing something mildly stupid when instructed to do so by a person with a clipboard, he acquiesced to read out a portion of the immense integer in an imbecilic made-up “seven-quinnelty-miliibong-quadro-trilli-trilli-thouswart” style, followed by the producer doing the same thing but with less fluent delivery, to the amused chortling of all. Aside from laughter at the general bigness of the number under discussion, Cooper covered most of the same points he was to reiterate the following morning, discussing the use of somewhat smaller primes in RSA encryption and the potential power of similar distributed-computing projects.

Chronic Aperiodical-hassler Matt Parker tweeted on Thursday night that he was due to appear on the World Service around the same time as Cooper – apparently, he did do an interview but it went out on World Service channels not accessible in the UK. Not, as we secretly suspected, that a strange person convinced him their living room was a BBC studio and got him to come in and talk about maths into a cardboard microphone.

A mere two hours later Cooper returned to the BBC studios for a grilling by Newsnight’s seventh-scariest presenter, Eddie Mair. To a montage of scrolling numbers and bits of miscellaneous computer kit, Mair gave a rather good potted history of the Mersenne primes (so far as is possible in forty-odd seconds), before Curtis Cooper popped up in a square over his left shoulder on the giant video-wall behind him, while what one can only assume were the digits of the quantity under discussion paraded past in the background. Either due to Newsnight’s computer memory being exhausted from storing the prime in its video-wall servers, or owing to some kind of satellite delay, the interview was plagued by a half-second lag, and Mair slightly annoyingly asked Cooper if he was “good with his tax return”, but at least got a good sense out of him of how the search actually got going and continues at his university.

The only BBC outfit to cover the story without wheeling out Cooper for a chat was Radio 4’s The World At One. Martha Kearney pulled the same trick as Today, prefacing their definition of a prime number with the disclaimer “as you will remember” to prevent the listener feeling like an idiot for not doing so. Kearney talked to Marcus du Sautoy, who was of course more interested in explaining a bit of the proper maths behind the patterns (or lack therof) in the primes than the big number itself. Kearney coined perhaps the best explanation of why people embark on this search, “for the sheer game of it”.

In the main, online coverage (which includes New Scientist, Plus Magazine and of course your own Aperiodical) has managed to correctly typeset that tricksy exponential notation and not make any inaccurate statements. Print media, in a post-Leveson world, of course had to tread carefully for fear of upsetting $2^{57,885,161}-1$’s libel lawyers. Of the mainstream newspapers, only the Telegraph and Independent seem to have even covered the story – of course, both gamely illustrated their coverage with a photo of an abacus. Coverage from The Guardian¹, The Times and bastion of science and maths coverage The Sun were all sadly absent.

The Telegraph got off to a bad start with its opening paragraph (“The number, expressed as 2 raised to the 57,885,161 power minus 1, can only be divided by itself and by 1, making it by far the largest prime number ever identified”) but the rest of the article is essentially correct, but slightly annoyingly refers to ‘Mersenne primes’ on more than one occasion without any attempt to define the concept. They also used the word ‘group’ when what they really meant was ‘set’, irritating truly committed mathematical pedants.

While The Telegraph wisely avoided attempting to type the number at all, The Independent fell foul of the gods of superscript and described the number as $257885161-1$ (eagle-eyed readers may be able to spot a divisor). They also used the magnificent “formula of $2p-1$”. Superscript issues aside, their article is very readable and covers all the main points well. It’s also accompanied by a “History of Prime Numbers” section to give some further maths background. Unfortunately it’s marred by a few errors, and while I choose to assume it at least looked pretty as a little fact-box in the dead-tree version, the grim reality of HTML has sadly sucked all the formatting life out of it.

While some of the coverage has been mildly annoying, it’s at least been on the whole largely accurate (Fox News excepted). Discussion continues as to how making a big deal out of this kind of discovery affects the public image of maths – does everyone (and a great deal more than that, if this tweet is to be believed) think it’s boring, and that mathematicians are just stamp collectors? Or is it just good to have maths in any context talked about on the radio and in the papers? The comments thread on this Slate article is full of people having a conversation about mathematics (best quote: “I wonder if a positive density of elliptic curves have Q-rank >0”) which then descends into quantum physics. Of course.

At least, thanks to Euclid, we know the media has another $\aleph_0$ attempts at getting this story right.

Listen Yourself:
BBC World Service – NewsHour (starts around 18:40)
BBC Radio 4 – Today (starts at 1:18:00)
BBC Radio 4 - The World At One (starts at 42:20)
Watch the segment on Newsnight

1. While the main Guardian newspaper had no article that we can find, the story was of course covered by lovely maths expert Alex Bellos (that is to say, an expert on maths who’s lovely, although I’m sure he knows a lot of lovely maths) and in what can only described as psychic journalism, Mersenne numbers were discussed in detail in a prescient and Numberphile-philic blog post by the Guardian’s GrrlScientist fully two days before the discovery was even made, let alone when the news broke.

In this short series of articles, I’m going to write about some mathematical questions we don’t know the answer to – which haven’t yet been proven or disproven. Hopefully you will find it interesting, and maybe someone will even be inspired to delve deeper and find the answers themselves.

Singmaster’s Conjecture

Pascal’s triangle is a thing of beauty – easy to explain and define, and yet underpinning the binomial coefficient. It is named after the French mathematician Blaise Pascal, although it was known and studied centuries earlier in many parts of the world.

To construct your own Pascal’s triangle is simple – start by writing the number $1$, and proceed by writing beneath each gap the sum of the two numbers above (using imagined zeroes at each end), which should give you the next row as a pair of ones, and then $1,2,1$ in the row below. This process can be continued forever, resulting in an infinite triangle of numbers getting wider and wider.

So what’s interesting about this? Well, if you’re looking to multiply out a bracketed pair of values, such as $(a+b)^n$, the coefficients of each term in the resulting expansion can be found in the appropriate row of this triangle. For example, $(a+b)^6$ multiplies out to give

\[ 1a^6 + 6a^5b + 15a^4b^2 + 20a^3b^3 + 15a^2b^4 + 6a^5b + 1b^6\]

Here the coefficients are just the row of the triangle starting $1,6$, which makes sense when you realise how they’re formed – as sums of the coefficients of smaller powers, each time a new $(a+b)$ is multiplied on.

Almost a side-fact about this triangle and its neat encoding of these coefficients is that the ‘choose’ function is very useful in maths. If you want to know the number of different ways to choose $n$ things from a pile of $m$ things – quite a common ask in combinatorics and probability – this can be written $n \choose m$, or ‘$n$ choose $m$’ (sometimes written $^nC_m$), which can be calculated using a horrible formula involving factorials, or it can be found as the $m$th entry in row $n$ of Pascal’s triangle like some kind of hilarious look-up table.

Since this beautiful triangular number-beast is so simple to define, and so nicely tied into a simple bit of mathematics, you’d think we’d have a handle on how it works. To some extent, we know lots of things – for instance, every row starts with a $1$, and then the number of that row, and ends similarly. We can also say for certain that no integer bigger than $1$ can occur in the triangle below its corresponding row; there are no fours below the fourth row, since the next row starts $1,5$ and everything proceeds by adding and making the numbers bigger. (Here, by the ‘fourth’ row, I mean the row starting $1,4$, so I must be numbering from $0$).

However, our knowledge has a hole in it. David Singmaster, mathematician and superstar of MathsJam conferences, has a conjecture named after him relating to Pascal’s Triangle, called Singmaster’s Conjecture. The conjecture states that we can find a number, $m$, so that no value occurs in Pascal’s triangle more than $m$ times. Obviously, this doesn’t include the number $1$, which occurs infinitely many times. While this conjecture seems obvious, given that no entry $n$ can occur in more than $n+1$ rows, it hasn’t been proved.

David Singmaster, still having fun with maths, at the recent MathsJam conference (photo: Jon Histead)

In his statement of the problem, in an article in The American Mathematical Monthly, Singmaster intimates that Paul Erdős agreed the conjecture is probably true but he suspected it would be very hard to prove. This is sometimes how it goes in maths – to get a verifiable, unquestionable proof of the truth of a fact is sometimes significantly more difficult than you would expect – even if the statement seems superficially obvious.

There have been various attempts by Singmaster himself, Erdős, and other mathematicians including Daniel Kane to determine the value of this upper bound, and there are formulae which give order approximations. Singmaster proved in 1971 that the number of occurrences of $a$ is $N(a)=O(\log a)$; that is, as you start to consider larger and larger numbers, the numbers of times they appear in the triangle do not get bigger very quickly. For Singmaster’s conjecture to be true, they would need to stop getting bigger at all after a certain point – in this notation, we’d need $N(a)=O(1)$.

The current best bound is due to Kane (2007) and is:

\[N(a) = O \left( \frac{(\log a)(\log\log\log a)}{(\log \log a)^3}\right)\]

this function grows even more slowly than Singmaster’s (but will still eventually become as large as you like, so doesn’t prove the conjecture). It’s also been shown that there must be infinitely many entries which occur six times. But no-one can say exactly what the most times any number will appear is – according to Wikipedia, Singmaster thinks it might be 10 or 12.

It’s also interesting how many facts remain unknown about multiplicities above 4 – the smallest number to appear 8 times is 3003, which occurs twice in its own row, twice in row 78 and twice in rows 14 and 15. But it’s also the only number known to appear 8 or more times! There are also no known integers which occur in the triangle exactly five or seven times. Certainly more could be discovered, if they exist, by lengthy computation – but the general results may continue to elude us.

Here’s a fun side-fact about Pascal’s triangle – if you draw a huge one (on a hexagonal grid, if you have one), and then circle/colour in all the numbers divisible by 2, you’ll get a Sierpinski triangle! This was attempted at a recent Manchester MathsJam, and while a negative correlation was discovered between number of drinks and colouring skill, we also attempted colouring in different subsets – numbers divisible by 3,4,5 and so on. It’s a nice activity and gives some pretty patterns – odd prime numbers have an especially interesting pattern provided your triangle is big enough.

Further reading

How often does an integer occur as a binomial coefficient? by David Singmaster (JSTOR, $12)

Improved bounds on the number of ways of expressing $t$ as a binomial coefficient by Daniel M. Kane

Repeated binomial coefficients and Fibonacci numbers by David Singmaster