Sadly, I received word in April 2018 that The Center for Talented Youth at Johns Hopkins University decided to cease publication of Imagine magazine. May/June ’18 was the final issue. That summer, I also left UIC to start a new job at The University of Maine. With the normal demands of the publication cycle gone, and a big mathematics education project awaiting me at UMaine, I haven’t been able to devote much time to puzzlemaking and updating the website.
But that doesn’t mean that Knossos Games is no more. I’ve kept the website up and running, because I knew how many people enjoy the puzzles here. There are still parts of the website that need to be updated. There are puzzles that were published in Imagine that I’ve never had time to post on the website. And there are puzzles that I had been saving up for Imagine that never saw the light of day.
Just as there always has been, there is still a tremendous amount of potential for Knossos Games ahead. I’m eternally grateful for the 25 years that my puzzle column appeared in print. I’m happy that it gets to continue to live on here.
A year before Imagine magazine began, back in 1992, my very first published puzzle was in the pages of the Study of Exceptional Talent Precollege Newsletter. Since Imagineceased publication a couple of years ago, the people at SET thought that their newest members might be unaware of Knossos Games. This past January, I spoke with Carol Blackburn, a research psychologist, SET staff member, founding editor of Imagine, and longtime enthusiast of my puzzle column. The following is reprinted from the SET Precollege Newsletter with their gracious permission.
SET alumnus Tim Boester has been making puzzles for nearly as long as he can remember. He began publishing puzzles in Imagine magazine when he was in high school, and continued to do so throughout college, graduate school, and into his professional life as an educational psychologist and mathematics professor. Puzzles by Tim appeared in all but six of the 125 issues of Imagine. You can visit Tim’s wonderful online archive—a treasure trove of puzzles—which includes thoughtful explanations of how to solve his puzzles at knossosgames.com. In this spread, we showcase just a few of the puzzles he created over the years, and hear about his journey from a boy who loved puzzles to a professor who studies how students learn advanced mathematical concepts.
In this 1994 yearbook photo from his senior year at Western Reserve Academy, Tim has some of his favorite puzzles ready to hand out at a math event.
When did you start making puzzles?
My interest in making puzzles started early. I don’t know why. I remember drawing my first puzzle in kindergarten—a giant maze on packing paper. The teacher hung it on the blackboard near the door, so that my classmates could try it. Discovering graph paper soon after was a revelation. I went through reams of it in elementary and middle school. I grew up making puzzles.
For years, pencil and paper were the only tools I had. Even today, I still start by jotting down notes and sketches on graph paper. It helps me clarify my preliminary thoughts about a new puzzle idea. However, using the computerized tools we have now has definitely influenced my process in creating puzzles. For example, being able to rapidly prototype puzzles and print out several copies enables me to test different versions quickly. I can explore many more options than I could before. At times I wonder how I was able to create some puzzles as a kid when I had to do everything by hand.
Clockwise from top left: 3D Wall mazes, Woven mazes, Node mazes, Triangular mazes
Your early puzzles include a variety of beautiful, complex mazes and spatial logic problems, like the one where the chess knight must jump across a chess board with missing spaces.
Many of my early puzzles were simply variations of puzzles I had seen and liked. Creating my own puzzles inspired by ones I enjoyed helped me better understand how puzzles work and how they are constructed.
Knight Jumping Islands – Puzzle 1
When did you start publishing your puzzles?
I became a member of SET in eighth grade, and the summer before my junior year, I sent some puzzles with my SET Update. You and your colleagues liked them and asked to publish one in the SET Precollege Newsletter. Then you asked to print one in the very first issue of Imagine magazine, on the Creative Minds Imagine page. Then another. By volume five of Imagine, I was a regular contributor with my own column, Knossos Games.
Where did the name Knossos Games come from?
I wanted a title that communicated the long history of puzzles, but also felt modern and fun. Knossos is the city on the island of Crete where King Minos, according to Greek mythology, built the labyrinth holding the Minotaur. Knossos Games hopefully evokes the notion of puzzles inspired by this tradition.
Your range as a puzzlemaker—the types of puzzles you created and their settings—really grew over time. I still remember when you sent us a puzzle inspired by something you learned in a biology class.
Yes, the professor was talking about how a particular cellular transport system works, and I thought, this could make a good puzzle. There were some channels that allowed molecules to move alone (uniport), others that required co-transport with another molecule (symport), and still others that required transport of another molecule in the opposite direction (antiport). It took some thinking before I decided how to use those channels in a puzzle, but eventually I came up with the right structure. Instead of having a traditional finish line, the goal of the puzzle was to use the channel-based passageways to move objects through the puzzle until all the objects had been moved to receptors scattered throughout the puzzle.
Puzzles are always based on rules. Coming up with new puzzle ideas means coming up with new sets of rules. So I’m constantly on the lookout for real-world situations that contain some inherent rules or structure that I could build upon or transform into a puzzle.
Greek Temple – Puzzle 4
That biology puzzle was the first of your “themed” puzzles, where your Knossos Games puzzle related in some way to the theme of the issue of Imagine in which the puzzle would appear. You came up with some wonderfully creative ideas, like the archaeology puzzles with the Indiana Jones-like booby-trapped squares in a maze.
As I became more experienced, I was ready to take on greater challenges and wanted to focus on creating puzzles that are unique. About ten years into creating puzzles for Imagine, I decided that I wanted each puzzle to relate to the theme of each issue. That self-imposed constraint was difficult but inspired some of my best puzzles. It was a real struggle at times to fit each issue, with topics as diverse as “energy” to “service and leadership” to “medicine.” I never would have thought to base a puzzle on fish ladders if I hadn’t been given the topic of “marine biology.”
Clockwise from top left: Wind Farm puzzles, Logic 19 – Park Cleanup, Pathogen puzzles, Fish Ladder puzzles
The archaeology puzzles were made for a history issue. I had to make a bunch of them because there were many ways to set up the rules, and I had to try out various rules and see what would work. Often, I don’t know before I create a puzzle what the solution will be. The final form of the puzzle emerges only after trying out different rules and combinations of rules, following them to their logical conclusion, and examining the results. Writing puzzles is a great way to practice and solidify your abilities in deductive reasoning.
Gerrymandering – Puzzle 1
What was one of your favorite puzzles?
One of my favorites was actually suggested by you, Carol. For a political issue, you suggested creating a puzzle about gerrymandering, and it was an intriguing suggestion. Here again, I knew the idea could be used to make a puzzle, but it took a while to figure out exactly how. I searched to see if puzzles based on gerrymandering had been made, and I found some, but they didn’t hinge on how gerrymandering works. I thought, they are missing the point of what a gerrymandering puzzle could be.
In the puzzle I finally created, the nefarious task is to gerrymander local districts so that your side wins a majority of districts (seats) even if it does not win the majority of votes. These puzzles hopefully enlighten solvers about how incredibly influential districting can be on election results. By doing the puzzle, the solver really sees how gerrymandering works, why it is so politically powerful, and why it needs to be stopped. Later I made a few other puzzles about real-world topics, where working through the puzzle gives you a better understanding of the way a system works, like the puzzles about where to position windmills in a wind farm to maximize the energy capture, or tracking the spread of a pathogen in a population.
Tim in his grad school office at The University of Wisconsin-Madison. On the wall behind him are Knossos Games puzzles and problem-solving diagrams related to his work in educational psychology.
Clockwise from top left: Logic 3 – Castaway Part 1, Logic 8 – Flight Plans, Logic 10 – Castaway Part 2, Logic 17 – Missing Instructions
You also created some fiendishly difficult logic puzzles.
The mathematical discipline of logic has existed since the time of Aristotle, but logic puzzles are a much more recent phenomenon. The mathematician Raymond Smullyan galvanized the field with his creation of “knights and knaves” logic puzzles, in which all knights tell the truth and all knaves lie. Given a set of statements from different people, one can deduce who is telling the truth and who isn’t.
At first I avoided making logic puzzles, because I didn’t think I had anything to add to the genre. Eventually, I came up with a “knights and knaves” puzzle inspired by the TV show Survivor. In my version, the first few people kicked off the island would always tell the truth, the middle people would tell one true statement and one false statement, and the last people would all lie. The puzzle presents you with a collection of statements by various people, and you have no idea which ones are true and which are false, and you have to logically sort through them. Logic puzzles are a place where I can ramp up the difficulty, and the Survivor puzzles (1, 2) are really tough. With an overwhelming number of clues, a big part of the challenge is just figuring out where to start.
Have you watched many people solve your puzzles? I wonder if you are sometimes surprised at the strategies they use.
I haven’t done as much of that as I would have liked. I had more opportunity when I was a graduate student in educational psychology. Back then, it was easy to find students willing to test-drive a puzzle. It was always interesting to see if my cleverly laid traps actually tripped them up, or if they just sailed on by, not taken in by the snares I had planned.
In my work, I certainly spend a lot of time watching my students solve math problems, and I read lots of professional papers about others who have watched students solve problems. I am always looking at how people approach problems. I’ve been doing this for a while, but I am still surprised sometimes at the thinking that people put in to solving a problem—right or wrong.
How does puzzlemaking relate to your career?
Some of my colleagues over the years haven’t known what to make of my puzzle hobby. To them, it doesn’t have a clear connection to my occupation. Sure, some of the puzzles are math-related and sort of learning-related, but to them it’s a weird remnant of my origin story.
Really, being an educational psychologist and making puzzles are two sides of the same coin. During my professional education, I learned a lot about how people solve problems. As a professor, I am trying to understand how people are thinking through mathematical concepts; and in my classroom I’m trying to use that knowledge to make those concepts as accessible as possible. Maybe not easy, but accessible. How can I help students navigate a tricky concept most successfully? What will be the most common pitfalls for them?
But when I’m designing a puzzle, I’m using all that knowledge to make things harder for the puzzle solver. I’m trying to understand how people are thinking, but then I’m trying to throw obstacles in their path. The same expertise that makes me a good educator also makes me a dastardly puzzlemaker.
I want to thank you for all the dastardly and delightful fun you created for us. Imagine would not have been the same without you and Knossos Games.
It was a pleasure. Puzzlemaking has always been an excellent creative outlet. And now that I am a teacher, it’s been fun to be able to write problems for Knossos Games that I could never put on a test because my students would kill me.
My mom loves Christmas decorations. She decorates every available space in the house with garlands, greens, candles, wreaths, and lights. Her navity scene was hand-carved in Germany and was a gift from my dad on their honeymoon. But nothing compares to her collection of Christmas tree ornaments.
Every Thanksgiving, dozens and dozens of boxes descend from the attic to decorate the Christmas trees. Yes, trees. We started with one in the living room, but with so many ornaments, she became a museum curator, only able to display a small fraction of the continuously growing collection on any one holiday. For years, my dad refused to put up a second tree, knowing this would be the path to extreme self-indulgent holiday cheer.
And he was right. When a neighbor brought over a used artificial tree one year, the barrier holding back unrelenting twinkle and sparkle had been breeched. She’s now up to four trees: a live tree we get every year from a local Christmas tree farm, a small driftwood tree for displaying ornaments made from natural materials, a half-size artificial tree for the small ornaments, and the aforementioned full-size artificial tree that stays up all year. (The excuse: it’s too difficult to take apart and haul into the attic. Sure.)
In the sprit of her unrelenting passion for Christmas tree ornaments, the latest holiday logic problem celebrates the yearly chaos of choosing which ornaments should go where. This puzzle was modified from one originally posted in 2005. It is an homage to the holiday-themed grid-based logic problems I made as a kid, but which have been entirely lost. I updated the puzzle text and clues (including changing some of the ornaments and tree locations), added new graphics and a detailed solution.
I started writing holiday-themed logic problems as a kid. I really liked solving grid-based logic problems, and once I figured out how they worked, I started creating my own. Unfortunately, I can’t find any copies of those original puzzles I made. But those early puzzles inspired me to start writing holiday-themed logic problems again once I got serious about the Knossos Games website in the mid-2000’s. As these puzzles aren’t published in the magazine (or anywhere else), they are a special bonus to my website visitors.
Even though my earliest logic problems were of the grid-based variety (both solving and creating), I have avoided that type of puzzle when writing for the magazine. In fact, I refrained from writing logic problems for the magazine altogether for the first five years. It was only when I found an interesting though flawed logic problem that I published my own improved version in the magazine. The following year I wrote the two-year garden logic problem and have written about one per year every year since.
When writing a logic problem, you want to situate or structure the puzzle using a natural order or pre-existing rules. This helps solvers initially engage with the puzzle, as they can utilize their prior knowledge to gain an entry point and start solving. The order of Santa’s reindeer, memorialized in story and song, made for an obvious choice. I used the L-shaped arrangements of the two-year garden logic problem as the basic structure for this new puzzle.
This puzzle was originally posted to an older version of the website in 2004, then resurrected as part of the newest version of the website in 2015. When I decided I wanted to restart the holiday logic problem tradition, I found a copy of the clues to this puzzle, but no solution! I had to go back and solve the puzzle myself to remember how it worked. I rewrote the entire prologue to the puzzle, utilizing actual proclamations found on the internet as a basis for the language used. The new version of the puzzle added graphics and a detailed solution.
• Two logic puzzles were specifically styled after Raymond Smullyan’s “knights and knaves” or “truth tellers and liars” puzzles. The first Castaway puzzle (based on the reality television show Survivor), adds people who sometimes lie and sometimes tell the truth. Smullyan calls these people “normals”. The second Castaway puzzle, not mentioned in the article, defines different ways in which people might lie based on whom they are speaking to. I have not found a similar Smullyan puzzle (although by no means have I read them all).
The article also includes a sidebar of puzzle books. I’ve listed those in a separate blog post if you’re interested in details and links.
Again, I’d like to thank Imagine and my editor, Melissa Hartman, for giving me the opportunity to write an article like this, in addition to my continued puzzle contributions. If you like the article and Knossos Games in general, please consider subscribing.
If you’re here because you read the article I wrote on logic and puzzlemaking in the recent issue of Imagine, you might want to check out the following:
I also wrote about the updates (part 1, part 2) of the logic puzzles for the most recent version of the web site.
Finally, here’s the blog entry that started it all, where I discuss how making puzzles and my job as an educational psychologist are really two sides to the same coin.
I find the entire concept of genetic code fascinating. I’ve taken a few biology classes over the years, and every time we discussed DNA, I marveled at how something could simultaneously be so simple yet so complex. Thus, I had wanted for a long time to make a puzzle that relied on DNA. I knew the opportunity for that puzzle had arrived in the summer of 2010, when I learned of the topic for one of the Volume 18 issues of Imagine: Biotechnology. The second puzzle was subsequently created in 2013, for the Frontiers in Medicine issue.
In the earliest brainstorming phases of creating a brand new type of puzzle, I usually start by writing down all the ideas that I have. No matter if they become part of the puzzle or not, I just want to make note of everything in my head, so that later thoughts or avenues I pursue don’t cloud my original ideas. Sometimes, those original ideas are changed significantly by the time the final puzzle is produced. (Sometimes the original ideas don’t produce anything of value whatsoever!) Remarkably, for the DNA transposition puzzles, much of my original ideas appear unchanged in the final puzzle.
Notes from July 22, 2010:
Going back to mazes with structure and rules1, the puzzle contains a set of connected paths. What governs which paths you can take at any intersection is a token that changes. (Instead of keeping track of this in the physical space, like in the Cell Wall Transport System puzzles, this puzzle uses an external item, more like the subway token puzzles2.)
You have a bit of DNA. Some intersections do nothing, but some rearrange bits of the DNA according to order. For example,
would move the first bit to between the third and fourth (making it the new third). So GATC would become ATGC.
Different paths have different restrictions. For example, a path could only let pass bits of DNA that have the sequence “GA”. The original piece of DNA could pass through this, but not the new rearranged piece (as it does not have “GA”).
The crucial idea of a “token” that you carried through the puzzle was essentially what made this puzzle different from my previous puzzles, and helped to clarify the instructions to others. The “intersections” became the bubbles in the final puzzle, while the “paths” became the connecting tubes. The diagram is meant to be an iconic representation of the prototypical chemistry lab apparatus.
The instructions took several passes, with the assistance of a few biologists called in by myself or my editor, adjusting the vocabulary to best fit what was happening in the puzzle. For example, I initially used the word translocation, which I found out typically refers to moving whole parts of a chromosome. Transposition is more appropriate when moving a shorter sequence or, in this case, individual nucleotides. Also, using the term DNA isn’t appropriate here, since DNA refers to the entire molecule, not a short sequence of nucleotides. The instructions therefore use the term genetic material, even though DNA is retained in the name of the puzzle.
The design of the puzzle took a while to finalize. I needed to visually communicate how each bubble transformed the genetic material.
Early attempts directly translated what I had in my notes, using numbers to show the rearrangement of nucleotides. These needed to be large to clearly display the numbers, but were too large and too cluttered for the rest of the puzzle. Thankfully, I hit upon the idea of completely eliminating the numbers and letting greyscale boxes denote the positions of the nucleotides.
This type of puzzle necessitates charting all possible paths through the problem space in order to ensure the designated solution is the shortest. In other words, every pairing of position and genetic token that is possible by moving through the puzzle must be examined. Because of the cyclical nature of the puzzle (repetitively rearranging the nucleotides and moving back to the same physical position in the puzzle), loops are possible (moving around the puzzle and returning to the same location with the same order of nucleotides). This challenge of representing a problem space with these loops was resolved by using one-way arrows.
Problem space for example DNA Transposition puzzle, shown above
Moving up each arrow loops back to a position that could have been achieved using fewer moves (sometimes utilizing a very different path through the puzzle), while the most direct solution is highlighted in blue.
The first Pathogen puzzle was created in November, 2008 for the public health issue (16.3 – Jan/Feb 2009) of Imagine, but for various reasons, never made it to the website around that time. When the magazine returned to the topic of public health (23.3 – Jan/Feb 2016), it not only made sense to design another Pathogen puzzle, but also to prepare both puzzles for the website.
Pathogen example: two days to spread, one person vaccinated
While I’ve forgotten some of the details about how I first came up with the idea of a puzzle based on a disease, I do recall that this was a time when bird flu (H5N1) was in the news. I knew immediately that a puzzle about public health was going to focus on, in some form, the mathematics of disease transmission. It did take some creative effort, however, to take those ideas and form a puzzle with them.
If you think about it, the structure of the puzzle scenario does not make a whole lot of sense. If you know who is infected, the contact chart, and how many days the pathogen has had to spread, it really strains credibility that you wouldn’t already know who patient zero is or (in particular) who is vaccinated against the pathogen. The more realistic problem is: knowing who is infected and some idea of the contact chart, figuring out who else might be infected. But I couldn’t figure out how to make a puzzle from that situation which was still well-defined and not trivial. In other words, if the contact chart is completely defined, then determining who would be infected is easy. If the contact chart is not completely defined, then it’s impossible to completely determine who would be infected. This mirrors reality, but isn’t very compelling as a puzzle, since puzzles are supposed to have solutions.
The Pathogen puzzles as created aren’t that difficult, since you can figure them out through brute force if necessary: trying out every infected person as patient zero and seeing what happens. There are faster ways to solve the puzzle, of course, but even each detailed solution basically makes good guesses as to whom patient zero might be, then simply tries them all. I needed to explore the sensible possibilities to create and test out each puzzle, and while there is typically a different method used to create versus solve one of my puzzles, in this case I couldn’t think of an alternative.
Testing each infected person as patient zero (solution bottom right)
Finally, I’ve met experts in a wide variety of fields through my years as a student and now as a professor, and those connections definitely come in handy when writing a new puzzle in an area which I have no expertise. The instructions were vetted by people both on my end and at the magazine, and their most important suggestion was to use “vaccinate” instead of “inoculate”, as the latter can mean to deliberately introduce a pathogen but not necessarily to produce immunity (such as for a culture or as a treatment). Even though the scientific situation of the puzzle may not be realistic, I still want to get these educating details right.
