Happy birthday¹, Évariste Galois (25 Oct 1811- 31 May 1832)

“Liberté Toujours”

[Image conceived by Card Colm Mulcahy, realized by Dan Bascelli]

201 years old now, but you don’t look a day over 20.

One of a very select group. Never one of the pack.

Galois biographies

Wikipedia

Eric Weisstein’s World of Biography

MacTutor

1. Yesterday. Sorry about the delay – the mgmt

This Sunday, 21^st October 2012, marks what would have been the 98th birthday of Martin Gardner, American man of letters and numbers, as well as logic, puzzles, magic and scepticism. I had the good fortune to know Martin in the last decade of his life, and a more gentle and modest man you could not find, completely disproportionate to the forceful and wide influence he wielded for over 50 years as a science and mathematics journalist of the highest calibre.

The first time I met Martin he fooled me by showing me a tall thin glass and getting me to agree that its height exceeded its circumference, when in fact it didn’t.

Martin Gardner wrote every book on the six shelves behind him in the above photo. He’s best known for the 300 columns on recreational mathematics which he wrote for Scientific American between the mid 1950s and the early 1980s, now available in 15 collections, and also in one place on a searchable CD-ROM from the MAA. Others know him for his Annotated Alice (1961), which remains his best seller, with over half a million copies shifted.

Every autumn, in and around October, there are worldwide Celebration of Mind events small and large, informal and formal, at which children and adults, amateur and professional mathematicians, puzzlers and magicians gather to celebrate the endless creativity of the human mind, inspired by Gardner’s extensive written legacy.

Over 50 Celebrations of Mind has already been held this year. So far, there are only two Celebration of Mind events listed on the site map in England, and none at all in Wales or Scotland. The good news is that there is still plenty of time for interested people to host one in the coming month, and we hope that many teachers and maths clubs in the UK and elsewhere will seize the opportunity to organise one this year. The range of topics one could explore is vast, and ideas can be generated by looking at any of the links here, and the CoM resources page.

As one example, there are hexaflexagons, the subject of the column that started Martin’s long run at Scientific American. Vi Hart of the Khan Academy, who first absorbed the Gardner influence through her father, mathematician and sculptor George Hart (himself a first generation fan of Martin’s), has launched a series of excellent videos on this dynamic paper folding activity, especially for this year’s Celebrations of Mind. The first video has had over 4 millions views in the past three weeks, and given rise to a Celebration of Mind sponsored Flexagon Party craze around the globe.

(don’t miss the second and third videos from Vi Hart about hexaflexagons)

That site has full instructions for how to run a flexagon making event for participants of all ages. More mathematically mature readers may enjoy a special issue of the College Mathematics Journal from earlier this year, entirely devoted to mathematics inspired by Gardner.

Surprisingly, but of great importance, we note that Martin had no formal training in his field — indeed he never took a mathematics class at university — and he didn’t get into his stride until he was middle-aged. The lesson is clear, and it’s one that shines through his top notch and seductive writing: anyone can have fun with mathematics and rationality and it’s all within reach regardless of age or credentials.

Rather than asking “Is there a doctor in the house?” we should ask “Is there a curious person in the room?” and go from there. Are there more curious people in Britain and throughout the world who’d like to share that with some others in the weeks ahead? If so, please pick a time and a place, gather some friends or pupils, and register a Celebration of Mind or Flexagon Party.

We leave you with a tasty treat: how can you cut this brownie into two identically shaped pieces with a single (not necessarily straight) cut?

Keep Reading

Gathering for Gardner page about Martin

Celebration of Mind

Flexagon Party

Martin’s first Scientific American column

College Mathematics Journal special Martin Gardner issue

Food for thought: savory treats for the mind from the Julia Child of mathematics and rationality

In praise of a deservedly popular mathematician

In 2011, a little over 55,000 Irish students overall, in a country with a population of 4.6 million, sat the Leaving Certificate in their final days of secondary education. This year, just under 54,000 school leavers are taking the Leaving, as it’s known. I hope they’ve studied hard, and wish them every success!

This year’s big increase in those going for the higher level papers is not a random event: for a 4-year trial period, those who achieve a grade D3 or better on the higher level maths exams will get 25 bonus points towards their all-important CAO, which determines what university they get admitted to. It’s all part of a government initiative to reverse a perceived decline in maths performance of the nation’s youth.

It’s also, in part, a throwback to a system that was in place when I took the Leaving Cert many moons ago; in those days we got “double points” (the exact number depending on how well you did) for Honours Maths at a time when there were two levels, Pass or Honours.  You had to pass one of these to enter university.  Failing Honours — even though you might have passed Pass had you taken it — would keep you from moving on to third level education.  Today, it’s not that different for those who are nervous about their chances, and it explains now, as then, why many of those who plan to sit the Honours or higher level examination papers settle on the day for the safer Pass/General/Ordinary level papers: it’s less risky.

Now, thanks to Project Maths, there are actually three streams/levels, and the situation above has been adjusted a little.

Last year’s Higher Level papers generated some controversy: even before they were unveiled, there were concerns about the decline in numbers already alluded to.

See the papers for yourself: Paper 1 and Paper 2 are available online. Solutions — and plenty of lively student commentary — are also online if you poke around.

Question 9.c on paper two is curiously “self-reflective” (from the point of view of student populations). It starts:

“The mean percentage mark for candidates in the 2010 Leaving Certificate Higher Level Mathematics examination was 67·0%, with a standard deviation of 10·4%. The suggestion that candidates who appealed their results have, on average, similar results to all other candidates, is being investigated….”

It is believed that these papers are the work of one person who has been writing them for a number of years and that, unlike end of year university examination papers in Ireland (and the UK), there are no external examiners who must approve them months ahead of time.

Regardless, a system such as this ensures that one has some idea of what incoming university students know about mathematics.  Every single one of them has had 12 years of maths without a break.  It simply isn’t optional. And there are no remedial courses such as precalculus, trigonometry or “college algebra” at the third level.

Furthermore, just as in my day, even the pass/general/ordinary level students have to do differential and integral calculus. Here are the 2010 papers to demonstrate that all Irish school leavers know some calculus:

Ordinary Level Paper 1 from 2010

Ordinary Level Paper 2 from 2010 (note Simpson’s rule!)

This is all in stark contrast to the situation in the USA, where I currently live and teach. There, there is essentially no guaranteed minimum level in mathematics that one can expect an incoming university student to have achieved. Some have not taken mathematics for several years before they show up at the gates of third level institutions. As might be expected, that leads to all sorts of issues. But that’s a topic for another day.

Several years ago it dawned on me that the number of possible ways to order or permute the cards in a standard deck of size $52$ was inconceivably large. Of course it was — and still is — $52!$. That’s easy enough to scribble down (or even surpass spectacularly) without understanding just how far we are from familiar territory.

Let’s start with something smaller: the number of possible ways to order or permute just the hearts is $13! = 6,\!227,\!020,\!800$. That’s about what the world population was in 2002. So back then if somebody could have made a list of all possible ways to arrange those $13$ cards in a row, there would have been enough people on the planet for everyone to get one such permutation.

Had a joker been thrown in too, it wouldn’t have worked out so well. Even today, with the population of the planet presumed to hover around 7 billion, there would have to be some sharing of the permutations on the list. In fact, since $14!$ is about 87 billion, it seems safe to predict that it will be a very long time indeed before the world’s population is large enough so that everyone gets just one such ordering.

Let’s put this in a musical context. It was also a decade ago, in the spring of 2002, that the Queens of the Stone Age recorded their Songs for the Deaf album. The standard release lists 13 tracks, but there is also a hidden 14th track at the end. They could have issued each person on earth their own personal copy, with their own personal track order, and also exhausted all of the possibilities in the process, assuming they still finished with that hidden track.

Adele ‘s recent 21 album, however, only has 11 tracks so, noting that $11! = 39,\!916,\!800$, she’d have had to settle for Poland or California if she wanted to achieve the same effect on both counts.

That’s something to think about it the next time you hit shuffle on your favorite music player.

The number of possible ways to order all the red cards in a deck is $26!$ which is about $4 \times 10^{26}$. How big is that? It’s certainly bigger than the number of grains of sand in Brighton, or Britain, or all of the beaches on earth for that matter.

You can be 100% sure that the compilers of the 26-track early Kinks set didn’t actually consider all possible track orders. To do so would have required making a list four times as long as a list of $10^{26}$ items. There simply isn’t enough paper, or computer memory. As anyone who has even compiled such a songlist knows, they probably decided on openers and closers and used something like chronological order for the rest.

Now consider taking out the four Aces from a deck, and putting the remaining hearts together in some order on the left followed by the other 26 cards in some order on the right. That can be done in over $10^{50}$ ways which exceeds the number of atoms on Earth.

As for playing with the full deck, note that $52!$ is about $8 \times 10^{67}$, which in the great scheme of things isn’t so far from $10^{80}$, the current estimate for the number of atoms in the universe.

Needless to say, nobody’s ever explicitly considered all of the options here either. Likewise, “the people in the office on the afternoon of Friday, September 3rd” over at Sub Pop, when coming up with the precise order for The Sub Pop List of the Top 52 Tracks Sub Pop Released in the ’90s.

What does this all mean? Well, if you were to hit “shuffle” (not allowing repeats) with a playlist consisting of those 52 tracks, it could be argued that you’d almost certainly hear a set of music that nobody has ever heard before. The same applies to the 52-track version of The World of Nat King Cole.

For a deck of cards, it means that there are far more shuffled states than have ever been written down. Likewise, the totality of all deck orders that have ever been achieved with actual decks in the history of the world is a very thin set within the set of all possible deck orders.

A well-shuffled deck, such as the one displayed here, which is far from being in any “obvious” or recognisable order, is probably unique in the sense that nobody else has ever come up with it before.¹

As we’ve been saying all along, you too can be unique. Just keep shuffling. You’ll get there.

The other take-home lesson today is that you can shuffle till the cows come home, but you’ll still miss the vast majority of the possibilities. Or as Hamlet once said, “There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.”

1. I recall just using an order that resulted from numerous shuffles of an already seemingly well-jumbled deck

Let’s start at the beginning.

My Plus magazine puzzle from March asks “Which gives a tighter fit: a square peg in a round hole or a round peg in a square hole?” By “tighter” we mean that a higher proportion of the hole is occupied by the peg.

Note that everything is scalable when considering fit: think of shrinking or expanding the images above with a photocopier; it’s only the ratios of areas that are important, not the specific measurements.

This question has been asked and answered many times before, and in generalised form too (more on that below).

One could also ask which is a better fit on the internet, by comparing Google hits for “round peg in a square hole” versus “square peg in a round hole”. The latter has been gaining popularity for over a century, at least since Edward Bulwer Lytton wrote

You … insist upon forcing a square peg into a round hole, because in a round hole you, being a round peg, feel tight and comfortable. Now I call that irrational.

Have those who use this phrase considered the alternative above, and done the maths? Probably not.

Ponder this related puzzle.

What portion of the large square does the small square cover?

The answer to that one is “one half” — just imagine that the inner square is rotated $45^{\circ}$ and it becomes pretty clear, without any computation at all. This observation allows us to answer our question too.

Assume that the larger square in the triple nested image above has width $2$, and hence area $4$. Upon reflection—and rotation, as just suggested—the smaller inside square must have area $2$ (an application of Pythagoras also confirms this). The circle has area $\pi$ of course.

So the square peg in a round hole fit is $\frac{2}{\pi}$, and the round peg in a square hole fit is $\frac{\pi}{4}$.

Since $\pi > 3$ it follows that $\pi^2 > 8$, and hence $\pi > \frac{8}{\pi}$. Dividing by $4$ yields $\frac{\pi}{4} > \frac{2}{\pi}$.

Hence, a round peg in a square hole is a better fit than a square peg in a round hole!

As decimal approximations, we are comparing $0.7854$ with $0.6366$: a round peg fills up about 78.54% of a square hole, whereas a square peg only fills up about 63.66% of a round hole.

Maybe that’s why we sometimes see round tins in rectangular boxes, but square bottles such as those pictured on the right are rarely packed in cylindrical boxes, since the latter is a less economical use of space.

If we’re going to move up a little in dimension, from $2$, let’s go all the way to $3$, and ask:

Is a sphere in a cube also a better fit than a cube in a sphere? Can you do the appropriate computations in this case? Does it help to consider the 3D analogue of the nested image above, this time of a cube in a sphere in a cube, and then rotate the inner cube?

It’s fun to envision a sphere starting inside a cube and then expanding until it perfectly encloses the cube.

It turns out that we are comparing $\frac{\pi}{6}$ (52.36% filled) and $\frac{2}{\pi \sqrt{3}}$ ( 36.76% filled)

What about in dimension $4$? That’s a little harder for most of us to visualise, unless you have the 4D instincts of somebody like geometer Tom Banchoff! Is a $4$-ball a better fit in a $4$-cube than a $4$-cube in a $4$-ball?

Here, it turns out that we are comparing $\frac{\pi^2}{32}$ (30.84% filled) and $\frac{2}{\pi^2}$ ( 20.26% filled)

Notice that we are getting poorer fits in both cases as the dimension goes up. It gets worse (before it gets weird).

How is your sixth sense? In 6D we are comparing $\frac{\pi^3}{384}$ (8.07% filled) and $\frac{16}{9 \pi^3}$ ($5.73% filled)

In 8D it drops to $\frac{\pi^4}{6144}$ (1.59%) versus $\frac{3}{2 \pi^4}$ (1.54%), and by 10D, it’s $\frac{\pi^5}{122880}$ (0.025%) versus $\frac{768}{625 \pi^5}$ (0.04%).

The pattern of poorer performance persists. But wait, the fit facts have flipped: apparently, when $n = 10$, an $n$-sphere in an $n$-cube is a worse fit than an $n$-cube in an $n$-sphere!

Actually, the flip took place before the tenth dimension.

David Singmaster (now a London resident) examined this in detail half a century ago, computing the appropriate ratios of hypervolumes, and published his results in a November 1964 paper called “On Round Pegs in Square Holes and Square Pegs in Round Holes” in Mathematics Magazine, 37, 335-339, which is also available in the recent MAA/Cambridge book “Harmony of the World — 75 Years of Mathematics Magazine“.

His conclusion was that an $n$-ball in an $n$-cube is indeed a better fit than an $n$-cube in an $n$-ball, for $n = 2, 3, 4, 5, 6, 7$ and $8$, but that the situation is reversed from dimension $9$ onwards.

There is a sense in which things flip as we cross a mysterious dimension around $8.13794$ (see OEIS sequence A127454).

As discussed by Singmaster and summarised in the MathWorld article on the question, the formulae to be compared (in a particular dimension) involve factorials, and hence can be generalised to non-whole numbers using the Gamma function. Those generalised hypervolumes agree for the specific dimension number between $8$ and $9$ mentioned above.

So the answer to the question “In what flipping dimension is a square peg in a round hole just as good as a round peg in a square hole?” is, “In dimension $8.13794$, more or less!”¹

As for the continually shrinking numbers representing quality of fit, an anonymous comment at the Plus magazine puzzle site pointedly remarks, “Roughly speaking as you increase the number of dimensions, more and more of the volume of the hypercube is out near its corners – high dimension cubes are qualitatively more like hedgehogs than building blocks!”

Wolverhampton native (and unrepentant Hammers fan) Andew Worsey raises another interesting question in the comments following the Plus magazine puzzle:

“Is the same true for any regular $m$-gon, as opposed to the square, when you look at the inscribed and circumscribed circles, and what is the limiting behaviour?

Why not start with $m = 3$, that is to say the case of equilateral triangles and circles! Then try pentagons, hexagons, and so on. For any regular $m$-gon, the question is readily settled using trigonometry.

How about generalising that too, say to regular polyhedra and spheres in the third dimension? Try it for the four platonic solids other than the cube. For instance, does a sphere fit better in a dodecahedron than the other way around?

Moving up to higher dimensions, you need to consider the analogues of the Platonic solids, namely polytopes. In the fourth dimension, there are five non-hypercube polytopes to consider, but from dimension five on, there are only two possible non-hypercube polytopes.

Incidentally, it’s not just $n$-balls and $n$-cubes that surprise (by reversing roles) starting in dimension $9$. In the “Spheres and Hyperspheres” chapter of Martin Gardner’s Mathematical Circus (MAA, 1981) there’s a discussion of several strange phenomena which occur for the first time in dimension 9, including the one above and also a sphere packing paradox of Leo Moser’s. (Some of this is reproduced in the Gardner compendium “The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems“.)

One can of course become beautifully befuddled in lower and much more familiar dimensions. We leave you with the following problem, which Gardner observed is equivalent to a celestial conundrum over which Isaac Newton and Oxford astronomer David Gregory argued circa 1694, and which was not definitively settled for another 180 years:

Can 13 paper circles, each covering a 60-degree arc of a great circle on a sphere, be pasted on that sphere without overlapping?

1. Another flipping question—which is of peripheral interest when coding with playing cards—namely “for what number n between $3$ and $4$ does $n!$ first pass out $2^n$ ?” was only noted at the On-Line Encyclopedia of Integer Sequences quite recently, by this author.

Sylvester!

No, not James Joseph Sylvester, the English mathematician initially denied recognition and university degrees in his home country on account of his Jewish background (Trinity College in Dublin obliged with BA and MA in 1841), who had a big impact on the history of mathematics in the USA before finishing his career at Oxford.

We refer to a much earlier Sylvester, known as Sylvester II. Confused?

Born Gerbert d’Aurillac in France, this fellow became Pope Sylvester II in 999, and died this weekend past in 1003 (clearly not in his prime, arguably that had last been in 997)

See “A mathematician who became Pope“ for more on this fascinating man’s life.

Cantor’s diagonal argument from 1891 was truly revolutionary: an ingenious way to demonstrate that no matter what proposed list of all real numbers (or, say, just those between $0$ and $1$) is put forth, it’s easy to find a number which is definitely missing from the list.¹

In a nutshell, Cantor was the first to show that some infinities are bigger than others.

Cantor’s diagonalisation argument for the reals is watertight, and has proved to be a model of elegance and simplicity in the century plus that has passed since it first appeared.

That didn’t stop engineer William Dilworth publishing A correction in set theory, in which he refutes Cantor’s argument, in the Transactions of the Wisconsin Academy of Sciences in 1974.

Nor, when mathematician Underwood Dudley included Dilworth in his book Mathematical Cranks, did it stop him suing Dudley and his publishers in Milwaukee, Wisconsin, in 1995. You can read about that on Justia.com.

It says, “The complaint alleges that because Dilworth is not a professional mathematician he finds it very difficult to get his articles on mathematics published and being labeled a “crank” will create an additional obstacle.”

Luckily for sanity, “The district judge granted the motion to dismiss on the ground that the word “crank” is incapable of being defamatory; it is mere “rhetorical hyperbole.” ”

Woody Dudley adds,

After Dilworth saw himself in Mathematical Cranks he tried to get in touch with me but I, in keeping with best cranks policy, replied once and once only. Not one to be put off, one day he showed up in my town looking for me. In an amazing coincidence, the wife of my department chair overheard his asking someone about me and, being a nice person, took him to her husband who was able to get rid of him.

If I hadn’t been so standoffish, he might not have decided to sue me, the MAA, and my school for damages. He started in federal court in Wisconsin, where his case was thrown out. My respect for lawyers went up, first because he represented himself — I assume because he could find no lawyer to take the case — and second because the MAA lawyers (the MAA has legal insurance for this sort of thing) were able to find all sorts of precedents where people called other people awful things in print, up to and including “lazy, stupid, crap-shooting, chicken-stealing idiot”, without being held liable for damages.

He then appealed, as was his right, to the Seventh Circuit Court of Appeals where Judge Posner, a smart man, threw him out again. Rather than go to the Supreme Court, he sued again, this time in Wisconsin state court. He lost again, and was made to pay the defendants’ legal expenses. That put a stop to the suing.

Nevertheless, he continued to send me letters. Why he thought I’d answer them is a mystery and I ended by throwing them away unread. A few years ago he died, I think up in his 80s, probably thinking to the end that he was right.

Dilworth wasn’t the first person who felt that his entanglements with, and pronouncements on, infinite sets led to a loss of status. According to Wikipedia, Georg Cantor himself, back in 1904,

was outraged and agitated by a paper presented by Julius König at the Third International Congress of Mathematicians. The paper attempted to prove that the basic tenets of transfinite set theory were false. (König is now remembered as having only pointed out that some sets cannot be well-ordered, in disagreement with Cantor.) Since the paper had been read in front of his daughters and colleagues, Cantor perceived himself as having been publicly humiliated. Although Ernst Zermelo demonstrated less than a day later that König’s proof had failed, Cantor remained shaken, even momentarily questioning God.

(Georg Cantor: his mathematics and philosophy of the infinite, by Joseph W. Dauben , Boston: Harvard University Press, 1979. page. 248)

1. One has to pay close attention to realise that the same proof doesn’t also establish that the rationals are uncountable, bearing in mind that the Cantor pairing function shows that the rationals most certainly are countable. See http://en.wikipedia.org/wiki/Countable_set

1. $60\%$ of students who study Chinese are artistic and love poetry.
2. $20\%$ of students who study Business are artistic and love poetry, and
3. Only about $1\%$ of students study Chinese, whereas about $15\%$ of students study Business.

Thus, out of every $1000$ students, there are $10$ studying Chinese, of whom $6$ are artistic and love poetry, and also there are $150$ studying Business, of whom $30$ are artistic and love poetry.

So if a student is artistic and loves poetry, it’s $5$ times more likely she’s studying Business than Chinese.

So much for preconceptions (and “correlation”).

Each prime is represented (as a square) by its own colour, and luckily there’s an infinite number of both. Composites are represented by squares composed of collections of smaller squares or rectangles of appropriate colours.

She has arranged the natural numbers in columns of width ten. Interesting geometric and visual patterns emerge, and on the other side she’s knitted a version with eight to a column, which makes it easier to work in Octal.

As Sondra says, “One of the cool things about this sweater is that it works in any language and on any planet!!!”

Thanks to Ivars Peterson (on Twitter at‏ @mathtourist) for the pointer.

The standard argument goes like this:

\[ \begin{align}
\operatorname{Prob}(\textrm{at least one match}) &= 1 - \operatorname{Prob}(\textrm{no match}) \\
&= 1 - \operatorname{Prob}(\textrm{all 23 birthdays are different}) \\
&= 1 - \frac{365}{365} \times \frac{364}{365} \times \frac{363}{365} \times \dots{} \times \frac{343}{365}
\end{align} \]

which comes out a tad over 0.5, or 50%.

The same assumptions and arguments show that with 60 people selected at random one is over 99% likely (“almost certain”) to get a match.

The MathWorld page on the birthday problem mentions that with only 14 people selected at random, one has a (slightly) greater than 50% chance of getting an “almost match” — namely birthdays within a day of each other.

About twenty years ago, Persi Diaconis & Susan Holmes showed that if one takes into account that in modern times a significant proportion of births are induced, and tend not to be scheduled at weekends, then the resulting Mon-Fri clustering throws off the uniform distribution. This “lumpy logic” leads to an even higher chance of birthday coincidences.

For instance, with merely 16 or 17 people selected at random, one has a (slightly) greater than 50% chance of getting a birthday match under this distribution. And the more the merrier: with 25 people the chance of getting a match is notably higher than it was under the unrealistic uniform distribution assumption.

Now let’s switch attention from birthdays to cards, namely to a standard deck of 52, which has been thoroughly shuffled.

How many cards need we pick at random so that there is a greater than 50% chance of getting at least one pair with the same value?

Once again, let’s turn things around, and focus on the chances of there being no match.

If $k$ cards are picked at random from a full deck, where $3 \leq k \leq 13$, then since there are four cards of each value, it follows that:

\[ \operatorname{Prob}(\textrm{at least one match}) = \large 1 - \frac{52}{52} \times \frac{48}{51} \times \frac{44}{50} \times \dots{} \times \frac{52-4(k+1)}{52-k+1}. \]

Plugging in various values for $k$, we find that we need to pick at least 6 cards to be at least 50% sure of a card value match. Given 8 or 9 cards, there is a high probability (89% or 95%) of a match, and with 10 cards, it’s very likely indeed (a 98% chance) to occur.

Try it out with ten randomly chosen cards; it’s amazing how often one gets a lot more than a matching pair. Pairs of pairs, or even triple pairs, or three of a kind, are far from unusual.

We can take advantage of this card match principle in two ways, using different procedures, when dividing up 10 random cards into two poker hands in seemingly fair ways.

That’s exactly what we did in the June 2006 Card Colm at MAA Online: Better Poker Hands Guaranteed

Enjoy!