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.”

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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!

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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.”

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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.

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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.

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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!)

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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….

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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.

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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!

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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.

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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:

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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.

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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.

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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.

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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.

LangtonsAnt
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).

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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!