So I and my colleagues created three levels of contests — Beginning, Intermediate, and Advanced — since it seemed that it would be difficult to create a single contest that everyone could enjoy. But I did include three problems that were the same at *every* level, so all participants could talk about some aspect of the contests with each other.

Participants had a few days to get as many answers as they could, and we even had books for prizes! Many remarked how much they enjoyed working out these puzzles.

Now this conference took place before I started writing my blog. I have written several similar contests over the years for various audiences, and so I thought it would be nice to share some of my favorite puzzles from the contests with you. And so *The Puzzle Archives* are born!

First, I’ll share the three puzzles common to all three contests. I needed to create some puzzles which were fun, and didn’t require any specialized mathematical knowledge. As I’m a fan of cryptarithms and the conference took place in Denver, I created the following puzzle. Here, no letter stands for the digit “0.”

For the next puzzle, all you need to do is complete the magic square using the even numbers from 2 to 32. Each row, column, and diagonal should add up to the same number. There are two solutions to this puzzle — and so you need to find them both!

And of course, I had to include one of my favorite types of puzzles, a CrossNumber puzzle. Remember, no entry in a CrossNumber puzzle can begin with “0.”

I also included a few geometry problems, staples of *any* math contest. For the first one, you need to find the area of the *smallest* circle you could fit the following figure into. Both triangles are equilateral; the smaller has side length 1 and the larger has side length 2.

And for the second one, you need to find the radius of the larger circle. You are given that the smaller circle has a diameter of 2 units, and the sides of the square are 2 units long. Moreover, the smaller circle is tangent to the square at the midpoint of its top edge, and is also tangent to the larger circle.

The last two problems I’ll share from this contest are number puzzles. The first is a word problem, which I’ll include verbatim from the contest itself.

Tom and Jerry each have a bag of marbles. Tom says, “Hey, Jerry. I have four different colors of marbles in my bag. And the number of each is a different perfect square!” Jerry says, “Wow, Tom! I have four different colors of marbles, too, but the number of each of mine is a different perfect

cube!”If Tom and Jerry have the

sametotal number of marbles, what is the least number of marbles they can have?

And finally, another cryptarithm, but with a twist. In the following multiplication problem, *F,* *I,* *N,* and *D* represent different digits, and the *x*‘s can represent *any* digit. Your job is to find the number *F I N D. *(And yes, you have enough information to solve the puzzle!)

Happy solving! You can read more to see the solutions; I didn’t want to just put them at the bottom in case you accidentally saw any answers. I hope you enjoy this new thread!

(Note: The *FIND* puzzle was from a collection of problems shared by a colleague. The first geometry problem may have come from elsewhere, but after four years, I can’t quite remember….)

The simplest entry point into solving the first cryptarithm is to look at the fifth column. D would have to be 0 or 9, but it can’t be 0 since we are given there are no zeroes in the puzzle. The final solution:

For the magic square puzzle, you need to first add the even numbers from 2 to 32, which is 272. Since the four rows use all 16 numbers, each row, column, and diagonal sum to 272/4, or 68. The two solutions (which differ only by interchanging the middle two columns):

The solution to the CrossNumber puzzle:

Here is a diagram illustrating the solution to the first geometry puzzle. The essential idea is to use the geometry of the equilateral triangles to show that *ABC* is a right triangle, meaning that *AC* is a diameter of the circle. Since *AC* is then the area of the circle is

The second geometry problem is a little trickier. The diagram below will help, where *q* is the center of the larger circle. If we say the length of *pq* is 2 + *x,* then so is the length of *qr*, since they are both radii of the circle. Therefore, the length of *rs* must be 2 – *x.* Since the length of *sq* is 1, we may apply the Pythagorean theorem to triangle *qsr,* which gives *x* to be 1/8, so that the radius of the larger circle is 17/8.

As for marbles, Tom has 9 + 16 + 36 + 100 = 161 marbles, and Jerry has 1 + 8 + 27 + 125 = 161 marbles. Here, begin by looking for the smallest possible sum of four cubes (which won’t work); the next smallest sum is the one you want.

And for the final cryptarithm, start by observing that *N* must be 0, and because each partial product is only four digits long, this means that *F* must be fairly small. The answer is *FIND* = 3201.

Incidentally, these puzzles are the same ones I wrote about almost two years ago — hard to believe I’ve been blogging that long! So if you want to try them, you can look at Number Searches I and Number Searches II.

Since I had success with one round of puzzles, I thought I’d try again. This time, I wanted to try a few CrossNumber puzzles (which I wrote about on my third blog post). But as my audience was professional mathematicians and mathematics teachers, I wanted to try to come up with something a little more interesting than the puzzles in that post.

To my delight again, my new trio of puzzles was also accepted for publication! So I thought I’d share them with you. (And for those wondering, the editor does know I’m also blogging about these puzzles; very few of my followers are members of the MAA….)

Here is the first puzzle.

Answers are entered in the usual way, with the first digit of the number in the corresponding square, then going across or down as indicated. In the completed puzzle, every square must be filled.

I thought this was an interesting twist, since every answer is a *different* power of an integer. I included this as the “warmup” puzzle. It is not terribly difficult if you have some software (like Mathematica) where you can just print out all the different powers and see which ones fit. There are very few options, for example, for 3 Down.

The next puzzle is rather more challenging!

All the answers in this puzzle are perfect cubes with either three or four digits, and there are no empty squares in the completed puzzle. But you might be wondering — where are the Across and Down clues? Well, there aren’t any….

In this puzzle, the number of the clue tells you where the first digit of the number goes — or maybe the *last* digit. And there’s more — the number can be written either horizontally or vertically — that’s for you to decide! So, for example, if the answer to Clue 5 were “216,” there would be six different ways you could put it in the grid: the “2” can go in the square labelled 5, and the number can be written up, down, or to the left. Or the “6” can go in the square labelled 5, again with the same three options.

This makes for a more challenging puzzle. If you want to try it, here is some help. Let me give you a list of all the three- and four-digit cubes, along with their digit sum in parentheses: 125(8), 216(9), 343(10), 512(8), 729(18), 1000(1), 1331(8), 1728(18), 2197(19), 2744(17), 3375(18), 4096(19), 4913(17), 5832(18), 6859(28), 8000(8), 9261(18). And in case you’re wondering, a number which is a palindrome reads the same forwards and backwards, like 343 or 1331.

The third puzzle is a bit open-ended.

To solve it, you have to fill each square with a digit so that you can circle (word search style) as many two- and three-digit perfect squares as possible. In the example above, you would count both 144 *and* 441, but you would only count 49 once. You could also count the 25 as well as the 625.

I don’t actually know the solution to this puzzle. The best I could do was fill in the grid so I could circle 24 out of the 28 eligible perfect squares between 16 and 961. In my submission to MAA FOCUS, I ask if any solver can do better. Can you fit more than 24 perfect squares in the five-by-five grid? I’d like to know!

I’m very excited about my puzzles appearing in a magazine for mathematicians. I’m hoping to become a regular contributor to the Puzzle Page. It is fortunate that the editor likes the style of my puzzles — when the magazine gets a new editor, things may change. But until then, I’ll need to sharpen my wits to keep coming up with new puzzles!

]]>As usual, we began informally at 3:00, giving everyone plenty of time to make it through traffic and park. This time we had three speakers on the docket: Frank A. Farris, Phil Webster, and Roger Antonsen.

Frank started off the afternoon with a brief presentation, giving us a teaser for his upcoming March talk on *Vibrating Wallpaper*. Essentially, using the complex analysis of wave forms, he takes digital images and creates geometrical animations with musical accompaniment from them. A screenshot of a representative movie is shown below:

You can click here to watch the entire movie. More details will be forthcoming in the next installment of the Bay Area Mathematical Artists (though you can email him at ffarris@scu.edu if you have burning questions right now). Incidentally, the next meeting will be held at Frank’s institution, Santa Clara University; he has generously offered to host one Saturday this semester as we have several participants who drive up from the San Jose area.

Our second speaker was Phil Webster, whose talk was entitled *A Methodology for Creating Fractal Islamic Patterns.* Phil has been working with Islamic patterns for about five years now, and has come up with some remarkable images.

Here, you can see rings of 10 stars at various levels of magnification, all nested *very* carefully within each other. While it is fairly straightforward to iterate this process to create a fractal image, a difficulty arises when the number and size of rosettes at a given level of iteration are such that they start overlapping. At this point, a decision must be made about which rosettes to keep.

This decision involves both mathematical and artistic considerations, and is not always simple. One remark Phil hears fairly often is that he’s actually creating a model of the hyperbolic plane, but this is in fact *not* the case. Having sat down with him while he explained his methodology to me, I can attest to this fact. His work may be visually somewhat reminiscent of the hyperbolic plane, but the mathematics certainly is not.

Moreover, in addition to creating digital prints, Phil has also experimented with laser cutting Islamic patterns, as shown in the intricate pieces below.

If you would like to learn more about Phil’s Islamic fractal patterns, feel free to email him at phil@philwebsterdesign.com.

We ended with a talk by Roger Antonsen, *From Simplicity to Complexity.* Roger is giving a talk at the Museum of Mathematics in New York City next month, and wanted a chance to try out some ideas. He casually remarked he had 377 slides prepared, and indicated he needed to perhaps trim that number for his upcoming talk….

Roger remarked that as mathematicians, we know on a hands-on basis how very simple ideas can generate enormous complexity. But how do you communicate this idea to a general audience, many who are children? This is his challenge.

The idea of this “tryout” was that Roger would share some of his ideas with us, and we would give him some feedback on what we thought. One idea that was very popular with participants was a discussion of Langton’s ant. There are several websites you can visit — but to see a quick overview, visit the Wikipedia page.

The rules are simple (as you will already know if you googled it!). An ant starts on a grid consisting totally of white squares. If the ant is on a white square, it turns right a quarter-turn, moves ahead one square, and the square the ant was on turns to black. But if the ant is on a black square, it turns left a quarter-turn, moves one unit, and the square the ant was on turns to white.

It seems like a fairly simple set of rules. As the ant starts moving around, it seems to chaotically color the squares black and white in a random sort of pattern.

The image above shows the path of the ant after 11,000 steps (with the red pixel being the last step). Notice that the path has started to repeat, and continues to repeat forever!

Why? No one really knows. Yes, we can see that it actually *does* repeat, but only sometime after 10,000 *apparently* random steps. The behavior of this system has all of a sudden become very mysterious, without a clear indication of *why.*

If the rules for moving the ant always resulted in just random-looking behavior, perhaps no one would have looked any further. But there are so many surprises. Especially since there is no reason you have to stick to the rules above. As suggested in the Wikipedia article, you can add more colors, more rules, and even more ants….

For example, consider the set of rules in the following image. It should be relatively self-explanatory by now: there are four colors; if the ant is on a black square, turn right a quarter-turn and move forward one unit, then change the color of the square the ant was on to white; then continue (where green squares becomes black, in cyclic order).

This looks like a cardiod! And if you actually zoom in enough, you’ll see that this is the image after 500,000,000 iterations…though again, no one has the slightest idea *why* this happens. Why should a simple set of rules based on 90° rotations generate a cardioid, of all things?

From the simple to the complex! This was only one of literally dozens of topics Roger was able to elaborate on — and he illustrated each one he showed us with compelling images and animations. For more examples, please see his web page, or feel free to email him at rantonse@ifi.uio.no. You can also see the announcement for his MoMath talk here.

As usual, we went our for dinner afterwards, this time for Thai. It seems that no one wanted to leave — but some of the participants had a 90-minute drive ahead of them, so eventually we had to head home. Stay tuned for the summary of next month’s meeting, which will be at Santa Clara University!

]]>As I mentioned last week, I’ll begin with Magnus’ memoir on the Symmetry Congress (as you can see in the title of his memoir). His friend Chuck Stevens lived near where the Congress was held, and so met him at the airport and was his tour guide for the duration of his visit. (Note: the Society is still active — just google it!)

In this excerpt, Magnus remarks (start in the middle of line 4) that people who don’t know much about polyhedra always ask the same two questions: how long did it take you to make that model, and what do you do with them? I have had similar questions asked of me over the years as well; you just learn to be patient and hopefully enlighten…. Of course Magnus was always kind and generous with his responses.

You might be surprised by Magnus talking with a 10-year-old boy at the conference. Of course it may have been that Josh just happened to be staying at the same hotel, though that is unlikely since he was visiting relatives. More likely is that his aunt or uncle was a conference participant and brought him to the conference. I should remark that it is a common occurrence for a participant in an international conference to plan a family vacation around the trip, so you regularly see children of all ages at such conferences.

I direct your attention to last seven lines here. Magnus was perfectly happy to have his brandy, building polyhedra in an air-conditioned room rather than braving the summer heat to be “cultural.” For me, this emphasizes the simplicity of Magnus’ life. He did not need much to make him happy — some paper and glue, his building tools, his Bible, and perhaps a few other books on philosophy and theology. The quintessential minimalist life of a Benedictine monk.

Here, the second paragraph is interesting. In rereading it, I think I could imagine the exact expression on Magnus’ face when he heard “I’ll take it.” I know that this was a rare occurrence for Magnus. Perhaps it might be less so now; because of Magnus’ influence, as well as the explosion of computer graphics on the internet, people are generally more informed about polyhedra than they were in 1995.

Moreover, more and more high school geometry textbooks are moving away from exclusively two-column proofs, and some even have chapters devoted to the Platonic solids. I don’t think we’re at the point yet where “dodecahedron” is a household word…but we’re definitely moving, if slowly, closer to that point.

The final excerpt I’d like to share is from December 1995. I include this as another example of my collaboration with Magnus — our discussions of “perfect versions” of polyhedra. I’ll go into this example in more detail since it’s a bit easier to understand, but I note Magnus was not a fan of the adjective “perfect.” (And as a historical note, I had used the term “perfect version” and had also corresponded with Chuck Stevens, so Chuck must subsequently have talked to or corresponded with Magnus and used the term, and so Magnus thought Chuck came up with the term.)

I now agree, but have yet to come up with a better term. The basic idea is that some polyhedron models are *very* complicated to build. But for many of them, there are ways to make similar-looking polyhedra which are still aesthetically pleasing, but a bit easier to construct.

Let’s look at an example I mentioned a few weeks ago: the stellated truncated hexahedron, shown below.

Notice the blue regular octagrams. Now consider the octagrams shown here.

On the left is a regular octagram. If you draw a square around it, as shown, you divide the edges of the square in the ratio Notice that the octagram is divided into 17 smaller pieces by its edges.

However, if you start with a square and subdivide the edges into *equal* thirds, an interesting phenomenon occurs — there are four points where *three* edges intersect, resulting in a subdivision of the octagram into just 13 pieces.

You will note that this variation is not regular — the horizontal and vertical edges are not the same length as the diagonal edges. So any polyhedron with this octagram as a face would not be a uniform polyhedron.

However, it would be what Magnus referred to as a “variation” of a uniform polyhedron. So if we took the stellated truncated hexahedron, kept the planes containing the pink triangles just where they are, but slightly move the planes containing the blue octagrams toward the center, we would end up with the following polyhedron:

Note that the octagrams are now the octagram variations. Also notice how the pentagonal visible pink pieces are now rhombi, and the small blue square pieces are completely absent!

Such simplifications are typical when working with this kind of variation. Of course many polyhedra have such variations — but now isn’t time to go into further details. But these variations were among the polyhedra Magnus and I wrote about.

As I mentioned, there is little left of my correspondence with Magnus, since several years of emails have been lost. But I hope there is enough here to give you a sense of what Magnus was like as an individual, friend, and colleague. He never let his fame or reputation go to his head — all he was ever doing, as he saw it, was taking an idea already in the mind of God, and making it real.

He truly was humble, gentle, and kind — and of course a masterful geometer who significantly influenced the last few generations of polyhedron model builders. He will be missed.

]]>I left off with an excerpt from 10 June 1994; this next one is dated 18 June 1994.

This is another excerpt describing elements of our collaboration together; I point your attention to the first few sentences, where “stellations of two cores” are mentioned. Again, since the focus is on my work with Magnus, I will elaborate a bit on this idea without giving all the necessary background. (If you’re really curious, the internet has all the answers to your questions about stellations.)

Consider the following uniform polyhedron, called the *stellated truncated hexahedron.*

This name is perhaps a misnomer, since careful observation reveals that the octahedron bounded by the eight pink triangles in fact lies *entirely* inside the cube (hexahedron) bounded by the eight octagrams. So the octahedron does not actually truncate the cube.

Here is a partial figure from a paper I’m currently writing based on the notes referred to in the above excerpt.

Notice in d) how the red square truncates the black square to create a regular octagon. So this figure shows how the lines containing the edges of an octagon (analogous to faces of polyhedra in three dimensions) divide the plane into different regions.

But in h), we have a figure bounded by lines containing edges of two squares, but one lies entirely within the other. So we cannot choose a *single* polygon as generating this diagram. Therefore, we say this figure is generated by two cores — the two squares which do not intersect each other.

Of course there are many other polyhedra which may be thought of in this way, but hopefully these two examples illustrate this polyhedral thread in my collaboration with Magnus.

This excerpt is from November 1994. I include it as it illustrates how industrious Magnus was in building polyhedra. He would build polyhedron models and sell them at craft fairs to raise money for the Abbey or other related causes.

“Sales average only about a dollar an hour for my work.” Now 1994 was not all that long ago…a dollar an hour hardly seems fair for the work of a world-renowned model builder! But most people just see a few brightly colored bits of paper held together with a few drops of glue, and have no real idea about the mathematics behind the models or what it takes to build them.

So building polyhedra really was a labor of love for Magnus; he would have built them just to give them away. I recall one trip where I brought back three large trash bags filled with models which Magnus built but had no room to store. I kept some for myself, but then freely distributed them to mathematics teachers and students of all levels so they could take pleasure in holding and studying them. I hope some were inspired to build a few models on their own….

Again, Magnus stayed rather busy! Note that in 1995, he had already been a priest for 50 years. Also notice the reference to the ISIS Congress in Washington, D.C. Magnus wrote a fairly detailed “memoir” about his attendance at that conference. I’ll show you several excerpts from that memoir in next week’s post, as it comes rather later in the year (and so there isn’t room for those excerpts this week).

This is another example of a figure Magnus sent me as part of our ongoing collaboration. At the beginning of our correspondence, I had begun a textbook on polyhedra based on spherical trigonometry, which I used for a course I taught at the college level.

In particular, I was investigating the mathematics in *Spherical Models.* Recall that I wasn’t satisfied with numerical approximations — I wanted *exact* mathematical expressions for the angles used to build geodesic models based on polyhedra.

The above figure is based on one I used for designing spherical models where some of the faces of the polyhedron are pentagons. When projected onto a sphere, you needed to draw geodesics (great circles) connecting the vertices and calculate the individual segments in the spherical pentagram just created.

Now in the plane, when you join the vertices of a pentagon with diagonals, the diagonals are subdivided into the ratio φ : 1 : φ, where φ is the golden ratio. But the *angles* into which the diagonals of the spherical pentagram are subdivided isn’t a matter of calculating a simple ratio — there is a lot of trigonometry involved. Moreover, the ratio varies with the size of the original spherical pentagon.

In any case, here it not the place to discuss all the mathematical details. I just wanted to illustrate again one aspect of my correspondence with Magnus. I feel certain I must have sent him all my draft chapters, but likely he gave me his commentary when I visited, or perhaps somewhere in those emails that have been lost forever in some virtual black hole….

This last excerpt shows Magnus’ gentle nature. Perhaps you are not aware of the difference between the geometrical terms *trapezoid* and *trapezium* (see this article for a brief history) — but the meanings in the UK and US are swapped. What is a trapezoid here is a trapezium in the UK, and vice versa.

In any case, I made a reference to “Those crazy Brits!” when discussing this difference in nomenclature, and Magnus was sympathetic to their cause, as shown in the last few sentences of the excerpt. I can honestly say I cannot recall him *ever* saying a disparaging word about anyone (try counting the number of people you can say *that* about). He was always keen to understand all sides of an issue — and this always made philosophical and theological discussions so interesting.

I’ll stop with this comment today. Next week, we’ll begin by looking at Magnus’ memoir on the ISIS conference in Washington, D.C.

]]>I included this quote because of Magnus’ translation of Kepler: “Beyond doubt there exists in the mind of God the eternal form of all these shapes, which we call its truth.” Magnus had a very philosophical approach to mathematics, and always considered that his work was some manifestation of the Divine.

Is mathematics invented or discovered? This age-old and unresolved question has sparked much debate over the centuries. But for Magnus, the answer is clear: mathematics is discovered. The polyhedra he created all existed in the mind of God long before he reimagined them with a few pieces of paper and a little glue.

So Magnus’ vocation as a Benedictine monk strongly influenced how he thought about his life’s work. Having personally held and studied some of Magnus’ models, it is not a stretch to say that perhaps they were in fact touched by the Divine….

This excerpt again shows how very active Magnus was during the 1990s. But this particular letter is part of a thread in our correspondence where I was planning my first visit to see Magnus.

What made the trip a little more challenging is that I also wanted to visit Peter Messer in Wisconsin as well. Magnus introduced me to Peter as someone very interested in studying stellations of polyhedra, both from concrete and abstract viewpoints. (I also have a folder of correspondence from Peter, but that for another time!) Also, Peter was using Mathematica, which I was quite familiar with. So it seemed appropriate that we should meet.

Peter just recently retired, but was a dermatologist at the time, and hence had a busy professional schedule. So I was trying to juggle three schedules to arrange a series of visits — all through written correspondence! Now we would consider this approach somewhat anachronistic, but at the time, it was perfectly reasonable.

The title of my talk at the Joint Mathematics Meetings was *Working with Magnus Wenninger*. So in addition to excerpts which documented our getting to know each other, I also included excerpts which illustrated various aspects of our collaboration.

If you look back at the very first excerpt from last week’s post, you’ll notice the second paragraph begins “I’m in no hurry to get information about barycentric coordinates.” In my introductory letter to Magnus, I mentioned that I had begun work on finding coordinates of polyhedra using a barycentric coordinate system.

But over the next several months, I did send Magnus notes on barycentric coordinates, and he did study them. Now is not the time to go into a detailed discussion of barycentric coordinates — the point is that Magnus occasionally included diagrams in his correspondence to present his perspective on different aspects of our collaboration.

The first figure shows barycentric coordinates relative to a six-frequency triangle. The second figure shows how he would use this abstract idea to create a template which he would use to make what would be called a *six-frequency geodesic icosahedron. *Again, now is not the time to go into details, but I did want to include a brief description for those who are a little more familiar with polyhedra and geodesic structures.

Although, as I mentioned just a few moments ago, I began arranging my first visit with Magnus through letters, I must have finalized arrangements with him and Peter over the telephone. This excerpt from 10 June 1994 was the first letter since February which mentioned my visit; I must have realized that calling both Magnus and Peter was ultimately more efficient than writing them. I should mention that Magnus had difficulty hearing over the phone, and so we never had any substantial conversations on the telephone; our chats were confined to arranging logistics of visits or other such details.

I always enjoyed my visits with Magnus. I would stay in a guest room in the Abbey at St. John’s University in Collegeville, Minnesota — this was a spartan room with a simple bed, a small desk and chair, and I think maybe another chair to sit in…it’s been too many years to remember the exact details.

I’ll recall a typical day during one of our visits — although the individual visits tend to blur together. I’d say I made a half-dozen visits while I lived in the Midwest, including a time when Magnus visited me and I drove him to a conference at the University of Illinois Urbana-Champaign.

In any case, there was a common room for guests in the abbey for breakfast. Nothing too elaborate, but enough to get you going. I’d be on my own in the morning — working on polyhedral pursuits, or perhaps just walking around the beautiful campus. My visits were usually during the summer, so I was spared the cold of Minnesota winters.

Magnus would then find me around lunchtime. He had permission for me to have lunch and dinner with the other monks — a real treat, since this was not the usual protocol. But Magnus and I often had philosophical and theological discussions as well as conversations about polyhedra, and so I was welcome at meals. I can’t remember details, but the conversations were always quite stimulating. The Benedictines at St. John’s were rather liberal.

In the afternoon, we’d meet to discuss various aspects of polyhedra or perhaps work on some models. Often Magnus would be building while we talked.

Then I’d be on my own for a bit. Often I would go to one of the Offices in the Abbey church before dinner, and I’d meet Magnus after the service (the monks sat in their own section) and we’d walk down to dinner and more interesting conversation.

Our evening sessions were a bit more informal, and our discussions would often be more philosophical rather than polyhedral in nature. Magnus would have his nightly brandy — or perhaps a little Grand Marnier, since I would usually bring him a bottle when I visited.

Then it was early to bed for Magnus, although I was up a little later. I am not sure when the first Office was in the morning, but I am fairly certain I was always in bed at the time….

We must have made arrangements for future visits either by phone or email, since this is the only correspondence I have which includes any details of visits with Magnus. But they were wonderful times which I shall always remember fondly.

I’ll continue with excerpts from my correspondence with Magnus next week…so stay tuned!

]]>This is by no means meant to be a complete biography of Magnus’ life; rather, I hope you get to know Magnus a little through our particular correspondence. Of course everyone who knew Magnus had a slightly different experience.

I have about 65 letters in my folder of correspondence, which includes both letters from Magnus and my replies. I had to be selective in preparing my talk for the Joint Mathematics Meetings, and what follows are the excerpts I chose (in chronological order), along with some personal commentary.

I chose this excerpt for a few reasons. First, Tom Banchoff will also be giving a talk at the JMM in our session commemorating Magnus. But I also wanted to highlight the fact that Magnus, although already in his early seventies, was still very active. He was routinely invited to give talks and presentations, and also published papers occasionally.

And, as he indicated, when not otherwise occupied, he was *always* building models of one sort or another. And I really *do* mean always — even at conferences, he would often have a table set up where he would build models and discuss polyhedra with anyone interested enough to stop by and see what he was up to. Which was usually very many people….

This excerpt is from a letter dated 30 August 1993. It would take too long to set the stage for what this letter discusses, but I want to draw attention to the words “eye-balling” and “guessing” here.

Magnus’ focus was on *building* models, not on studying their precise mathematical properties. So if he didn’t actually know exact coordinates for a model, he would approximate, and build what he called a *variation* of the abstract mathematical model.

But here, Magnus is being somewhat humble — for his eye-balling and guessing is scaffolded on decades of model-building experience. It’s like giving a chef who graduated from Le Cordon Bleu a bowl of soup, say, and asking him or her to “guess” what spices were used. You’d get a lot more than just a guess….

This intuition about three-dimensional geometry was absolutely key to Magnus’ model-building success. If a spike on a model was too “pointy,” Magnus would simply design a variation with the features he wanted. And somehow, almost magically, they always turned out wonderfully.

This next excerpt is dated 13 September 1993. The important quote here is the last complete sentence. As I mentioned last week, Magnus is best known for his trio of books *Polyhedron Models,* *Spherical Models,* and *Dual Models.*

Magnus is not the first to have enumerated all the uniform polyhedra; this had been done by a few others earlier — Coxeter (et al.) published a complete enumeration in 1954, Sopov proved completeness of the list in 1970, and Magnus’ *Polyhedron Models* was published in 1971.

And even though *Polyhedron Models* was always the most popular of the three books, Magnus’ felt that *Dual Models* was more significant since it was the first time all the *duals* of the uniform polyhedra were published.

I also note his use of words: “more significant.” I don’t think Magnus would never had said “I am more proud of *Dual Models.” *He was not a prideful man, always giving credit to his God for his insights and abilities.

But I think significance here is all relative. In terms of popularizing polyhedra and model building, *Polyhedron Models* was certainly the most significant of the three books. But in terms of a contribution to the mathematics of polyhedral geometry, I would agree with Magnus that *Dual Models* was the most significant of the three.

Of course *Spherical Models* is quite wonderful, too…. But in this book, Magnus indicated how to build (or approximately build) a wide variety of spherical models. However, moving in this direction is much more difficult mathematically and in some sense broader than just considering the uniform polyhedra or their duals, since there are *many* ways to build a spherical model based on a given polyhedron. So there would be little hope of being so comprehensive when discussing spherical models.

I include this next excerpt since it is one of the few times Magnus has mentioned his own abilities in his letters — it also echoes the excerpt from 30 August 1993. So there is little more to say here, except to note that he mentions his intuitive abilities are derived from building over a thousand models, not to any inherent ability. Again, an instance of his humility.

In this excerpt, I point to the phrase, “…holding a *real* model in my hands is still my greatest thrill.” Those of us who knew Magnus absolutely knew he felt this way about model building.

Because of his extensive polyhedral network, he had connections with mathematicians who helped him with precise, computer-generated diagrams to help him make models for his books. But he was less interested in the details of how the diagrams were made than he was in diving right into building the models themselves.

He was also well aware that computer graphics were an excellent tool which could be used to discover new and wonderful polyhedra — changing a parameter here or there would result in a different polyhedron, and it is easy to look at parameterized series of polyhedra with a computer.

But, this was all in aid of model building, not an end in itself for Magnus. There are plenty of others who focus on the computer graphics — as it should be, since any polyhedral enthusiast will have their own particular areas of interest and expertise.

Only five excerpts, and it’s time to finish! I didn’t realize I had so much to say about these snippets of correspondence. But I hope you are beginning to have some insight into who Magnus was and what he accomplished. I’ll have still more to say as in the next installment of excerpts from my correspondence with Magnus.

]]>These books were important because they opened up the world of polyhedron model building to a much wider audience, giving detailed instructions on how to build dozens of models, as well as discussing the mathematics underlying polyhedra (although to a limited extent).

But perhaps equal to Fr. Wenninger’s model-building capabilities was his ability to connect polyhedron enthusiasts with one another. Because of his books, other model builders would correspond with him describing their particular interests, and he would connect them with others who had written to him with similar interests. In 1996, he created a mailing list where those interested could exchange ideas about all aspects of polyhedra. It was very active for several years, but because of the proliferation of sites about polyhedra in recent years, it is somewhat less so now. It is now maintained by Dr. Roman Mäder.

I was first introduced to Fr. Wenninger’s books by finding them among the stacks at the mathematics and science library at Carnegie Mellon. I don’t recall how many times I checked out *Polyhedron Models;* I was completely engrossed. I would page back and forth, over and over, looking for similarities between the various models and underlying geometrical patterns.

I began corresponding with Magnus in the summer of 1993. I still have all of our correspondence — I made sure to photocopy any letter I sent him so I would have a continuous record of our polyhedral conversations.

Sadly, Magnus Wenninger passed away last February at the age of 97. When I heard about this, I thought it would be fitting to organize an Invited Paper Session at the Joint Mathematics Meetings (on 12 January 2018 in San Diego) in his honor. I invited some of his colleagues from the polyhedron mailing list, and others as well.

This morning, I began reading through the letters exchanged with Magnus in preparation for a talk I will be giving in the session. Yes, this was in the day when people wrote *letters**,* and moreover, wrote them *by hand.* Amazing! Such an interesting treasure trove of ideas and thoughts.

For my talk, I am excerpting text from Magnus’ letters to me, taking pictures of the text, and using these excerpts as the main body of slides. I am not sure how many others had an extensive correspondence with Magnus, but I thought this would provide a unique glimpse into Magnus’ life.

And so begins this series of posts commemorating Fr. Magnus Wenninger. I’ll continue this post by giving an overview of our relationship. Then later, I’ll share with you some excerpts from his letters to me, as well as provide commentary when appropriate. It is fitting that he should be remembered; organizing the Invited Paper Session in his memory as well as writing about him in my blog will serve as my contribution.

I was simply fascinated by the beauty, intricacy — and to an extent, simplicity — of three-dimensional polyhedron models. But I was *also* a graduate student at Carnegie Mellon in mathematics, which was not insignificant.

And just what *was* the significance of being a graduate student? Well, I was being trained to think rigorously, mathematically. At the time, however, all the accessible books on polyhedral geometry were at a relatively elementary level.

What I mean is this. When you study a polyhedron, there are many metrical features evident, such as edge lengths and various angles; for example, the angles between two faces of a polyhedron (like the 90° angles between faces of a cube). And in many of these books, these lengths and angles were given — but in most cases, only in tables with approximations to enough significant figures necessary to build reasonably accurate models.

I started to wonder *how* all those numbers were calculated. I wasn’t satisfied with *approximate* results; I wanted *exact* results. Thus began my polyhedral self-education, some thirty years ago.

So I began playing around, and when I finished graduate school, mustered up the courage to write Magnus. I was rather intimidated at the time — I had just started learning about polyhedra, and he had published at least three books on the subject! But it turns out Magnus was unusually generous with his time and talent, and replied within a few weeks.

My introductory letter was dated 5 July 1993, and the last letter I have from Magnus bears the date 22 October 1997. I am quite sure that this is because we continued our correspondence online. However, virtually all of that correspondence is lost, since most of it was conducted through a university email address I no longer have access to.

There are a few lingering emails in my gmail account, since my correspondence with Magnus waned after I left my first university position. But in addition to maintaining an active correspondence with Magnus, I did visit him a number of times at St. John’s Abbey in Collegeville, MN. I would stay in a guest room at the Abbey, and join Magnus for lunch and dinner. Throughout the day, I would visit Magnus’ room where we would talk about polyhedra, or perhaps I would work on my own, or just take a walk around the campus.

My interactions with Magnus were absolutely inspirational. Looking back at notes from those years, I am amazed at how much I accomplished — and this at a time when computer graphics were much less sophisticated than they are now. I would often share results with Magnus in my letters, and he would provide his unique perspective on my current work. Even when he was critical, he was unfailingly kind.

So in my next post, I’ll begin sharing excerpts from Magnus’ letters to me. That is, after all, how you get to know a person — one interaction at a time. Hopefully you will be able to get a sense of the humble, brilliant, generous man Magnus was and continues to be for those who knew him well. His legacy lives on through us, as we strive to be for others who Magnus was for us.

]]>For a more complete description together with an example of how the game is played, you can look at the previous installment of Beguiling Games.

OK, now for the solution! It turns out that the *second* player can always win. Let’s see how. I found it easiest to think about a strategy by imagining the grid as being divided into quadrants, like this:

Now here is the important observation: the first player who fills in a second square in *any* of the quadrants loses, regardless of whether the squares are adjacent or diagonally situated. Of course there are other ways to lose, too — as with all these two-player games, there are multiple ways to analyze them.

So let’s get specific. Suppose the first player colors in square A1 (see the figure below). The second player then colors in the square labelled B1.

At this point, the red squares indicate all the places the first player *cannot* play without losing the next turn. So the first player must color in one of the two empty squares, after which the second player will color in the other one. So after two turns for each player, the board now looks like this:

So no matter what square the first player colors in next, one of the quadrants will contain two filled-in squares, and so the second player will win on the next move.

A similar strategy may be used no matter where the first player begins. Consider the first few moves in the following game. The first player colors in the square A1, and the second player colors in B1.

Again, the first player must avoid the red squares, or else the second player would win on the next turn. Whichever square the first player colors in next, the second player can always play “two away.” The result will be 1) the first player will not be able to win on the next turn, and 2) one square in every quadrant will be colored in. This means that the first player is forced to put a second square in one of the quadrants on the next move, meaning that the second player will win on the turn after that.

This is the simplest strategy I found for the second player. I would be happy to hear if some reader found an even simpler way to describe a winning strategy!

What about using other splotches? If the splotch contains too many squares, it is possible to force a draw. For example, given the splotch below, either player may force a draw simply by coloring in the four corners on their first four moves.

Interestingly, it is difficult to come up with a splotch where the *first* player has a winning strategy (other than a splotch which is just a single square, of course). The more squares included in the splotch, the more difficult the analysis. But for simpler splotches, it seems a clever division of the board allows the second player to win.

For example, consider the following square splotch.

Now divide the board into the following 2 x 1 regions, or dominoes:

Player two has a simple winning strategy. Whenever the first player fills in a square, the second player fills in the *other* square of the domino. It should be clear that the second player can *never* lose this way. The first player will eventually have to fill in a square directly above or below a filled-in domino, and when this happens, the second player wins on the next move.

A complete analysis of *Splotch!* is likely beyond reach. Just *counting* the number of possible splotches (up to rotation and reflection) would be a challenging task unless you wrote a computer program to exhaustively find them. Without rotations and reflections, there are 2^{16} = 65,536 possible subsets of 16 squares, and hence 65,535 splotches (since a splotch must include at least one square). So a computer program would be able to find them all relatively quickly. The interested reader is welcome to undertake such a task….

Here is another simple two-player game for you to think about, which I call *Scruffle.* It is played on a typical 3 x 3 Tic-Tac-Toe grid. Players alternate playing either a 1, 2, or 3 anywhere in the grid. A player wins when a number they place creates a column, row, or diagonal which contains a 1, 2, and 3 in any order.

There is one additional constraint: only three of each number may be placed in the grid. So once three 1’s (for example) are placed in the grid, no player may place another 1 anywhere in the grid. This is not an arbitrary constraint — you can show that the game *cannot* end in a draw with this condition. See if you can show this!

For the first puzzle, show that the first player has a winning strategy. This is not difficult; the simplest strategy I found involves the first player’s second turn involving playing the same number they played on the first turn.

A slightly more challenging puzzle is to require the first player to play a *different* number than the number they played on their first move. Does the first player still have a winning strategy? I’ll give you the solution in the next installment of Beguiling Games!

As we had been doing before, we began with a social half hour while waiting for everyone to show up. We then moved on to the more formal part of the afternoon.

There were three speakers originally slated to give presentations, but one had to cancel due to illness. Still, we had two very interesting talks.

The first talk, *Squircular Calculations,* was given by Chamberlain Fong. Chamberlain did speak at the inaugural September meeting, but wanted a chance to practice a new talk he will be giving at the Joint Mathematics Meetings in San Diego this upcoming January.

So what is a *squircle*? Let’s start with a well-known family of curves parameterized by *p* > 0:

When *p* = 2, this gives the usual equation for a circle of radius 1 centered at the origin. As *p* increases, this curve more and more closely approaches a square, and it is often said that “*p* = ∞” is in fact a square.

However, in Chamberlain’s opinion, the algebra becomes a bit unwieldy with this way of moving from a circle to a square. He prefers the following parameterization:

where *s* = 0 gives a circle, and the central portion of the curve when *s* = 1 is a square. As *s* varies continuously from 0 to 1, the central portion of this curve continuously transforms from a circle to a square. This parameterization was created by Manuel Fernandez Guasti; you can read his original paper here.

Chamberlain’s talk was about extending this idea in various ways into three dimensions. He showed images of squircular cylinders, squircular cones, etc., and also gave equations in three-dimensional Cartesian coordinates for all these surfaces. You can see some of the images in the title page of his presentation above. It was quite fascinating, and there were lots of questions for Chamberlain when his talk was finished. Feel free to email him at chamb3rlain@yahoo.com if you have further questions about squircles.

The second talk was given by Dan Bach (also a speaker at our inaugural meeting), entitled *Making Curfaces with Mathematica.* Yes, “curfaces,” not “surfaces”!

Dan took us through a tour of his *very* extensive library of Mathematica-generated images. He is fond of describing curves using parameters, and then changing the parameters over and over again to generate new images.

This is easy to do in Mathematica using the “Manipulate” command; below is a screen shot from Mathematica’s online documentation showing an example.

The parameter *n* is used in plotting a simple sine function — as you move the slider, the graph changes dynamically. Note that *any* numerical parameter may be experimented with in this way. Simply make a slider and watch how your image changes with the varying parameter.

So what are “curfaces”? Dan uses the term for images create by a family of closely related curves which, when graphed together, suggest a surface. As we see in the example above, the family of curves suggests a spiraling ribbon in which several brightly colored balls are nestled. Dan showed several more examples of this and discussed the process he used to create them. To see more examples, you can visit his website www.dansmath.com or email him at art@dansmath.com.

Once the talks were over, we had some time for puzzles! Earlier in the week, when I knew we were not going to have an overabundance of talks, I asked participants to bring some of their favorite puzzles so we could all have some fun after the talks. We were all intrigued with the wide variety of puzzles participants brought.

My dissection puzzle was actually quite popular — that is, until a few of the participants solved it!

You might recognize this from my recent blog post on geometrical dissections. The pieces above are arranged to make a square, but they may *also* be rearranged to make an irregular dodecagon. Some asked if I had any more copies of this puzzle, but unfortunately, I didn’t. Maybe I’ll have to start making some….

As has been our tradition, many of us went out to dinner afterwards. We went to our favorite nearby Indian buffet, and engaged in animated conversation. Interestingly, after talking a bit about mathematics and art, Chamberlain began entertaining us with his wide repertoire of word puzzles.

To give just one example, he asked us to come up with what he calls “mismisnomers.” Usually, the prefix “mis-” means to incorrectly take an action, as in “misspell.” But some words, like “misnomer,” begin with “mis-,” while the remainder of the word, “nomer” is not even a word! How many mismisnomers can you think of? This and similar amusing puzzles kept us going for quite a while, until it was finally time to head home for the evening.

So that’s all for the Bay Area Mathematical Artists in 2017. Stay tuned in 2018…our first meeting next year will be at the end of January, and I’ll be sure to let you know how it goes!

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