The Puzzle Archives, I

In going through some folders in my office the other day, I came across some sets of mathematics puzzles I wrote for a conference of the International Group for Mathematical Creativity and Giftedness in 2014.  Teachers of mathematics of all levels attended, from elementary school to university.  The organizing committee (which included me) thought it might be fun to have some mathematical activity that conference attendees could participate in.

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 same total 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….)
Continue reading The Puzzle Archives, I

More CrossNumber Puzzles

Last fall, I mentioned that while looking at the Puzzle Page of the FOCUS magazine published by the Mathematical Association of America, I thought to myself, “Hey, I write lots of puzzles.  Maybe some of mine can get published!”  So I submitted a few Number Search puzzles to the editor, and to my delight, she included them in the December/January issue.  Here’s the proof….


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!

Bay Area Mathematical Artists, V.

The Spring semester is now well underway!  This means it’s time for the Bay Area Mathematical Artists to begin meeting.  This weekend, we had our first meeting of 2018 at the University of San Francisco.

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 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

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.

2018-02-03 From Simplicity to Complexity.012

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.

From the Wikipedia commons, user Krwawobrody.

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).

2018-02-03 From Simplicity to Complexity 54.001

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  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!

Fr. Magnus Wenninger, O.S.B., V

This is the final installment about my correspondence with Magnus Wenninger.  I didn’t realize I had so much to say!  But I am glad to take the opportunity to share a bit about a friend and colleague who contributed so much to the revitalization of three-dimensional geometry in recent years.  Talk to anyone truly interested in polyhedra, and they will know of Magnus.

Excerpt from 14 August 1995.

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!)

Excerpt from 14 August 1995.

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.

Excerpt from 14 August 1995.

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.

Excerpt from 14 August 1995.

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.

Excerpt from 4 December 1995.

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 1:\sqrt2:1.  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.

Fr. Magnus Wenninger, O.S.B., IV

Last week I ended with a remembrance of my wonderful visits to see Magnus at St. John’s Abbey in Collegeville, MN.  These were certainly unique among my visits with other friends and mathematicians.

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

Excerpt from 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.

Excerpt from November 1994.

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….

Excerpt from 6 March 1995.

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).

Excerpt from 20 March 1995.

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….

Excerpt from 20 April 1995.

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.

Fr. Magnus Wenninger, O.S.B., III

I just returned yesterday from the Joint Mathematics Meetings in San Diego, where I gave my talk commemorating Fr. Magnus Wenninger.  Last week, I posted several excerpts from my correspondence with Magnus which I included in that talk.  Today, I’ll continue that discussion, again adding commentary as appropriate.

Excerpt from 3 February 1994.

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….

Excerpt from 27 February 1994.

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.

Excerpt from 3 March 1994.
Excerpt from 16 May 1994.

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.

Excerpt from 10 June 1994.

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!

Fr. Magnus Wenninger, O.S.B., II

In this next installment of my commemoration of Fr. Magnus Wenninger, I’ll excerpt various snippets from our correspondence together which illustrate Magnus’ unique perspective as well as highlight our collaboration together.

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.

Excerpt dated 24 August 1993.

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….

Excerpt dated 30 August 1993.

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.

Excerpt dated 13 September 1993.

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.

Excerpt dated 4 October 1993.

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.

Excerpt from 6 November 1993.

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.