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.
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.
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.
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.
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.
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.
The firstGreek Temple puzzle coincided with the first time that I tried to align the content of my puzzle column to the content of the magazine. Prior to the fall of 2004, I basically just created whatever puzzles I wanted to. Starting with Issue 12 of Imagine, I began creating puzzles that matched the theme of each issue. With the first issue being Archaeology & Paleobiology, this provided an opportunity to publish a new type of puzzle I had been working on for a while.
Unfortunately, back then I didn’t consistently keep very good notes about creating puzzles, so I don’t know exactly when I created the first Greek Temple puzzle. I do know that, prior to the fall of 2004, I had made an entire set of smaller Greek Temple puzzles based on the idea of linking the four possible state changes for the gateways (open, close, change, same) to the four possible options of moving between two tiles (alpha to alpha, alpha to beta, beta to alpha, beta to beta). While the graphical style of the puzzle has remained remarkably consistent over the years, I don’t remember how I came up with that original idea. The set of smaller puzzles has never been published, as they really belong together as a complete set; instead, each time a history or archeology issue comes around, I’ve chosen to create a new Greek Temple puzzle.
I did create a backstory for the puzzles to help me with some of the design details: A couple of archeologists have recently unearthed these ancient yet pristinely preserved structures. They accidentally realize that, with the introduction of a source of water, stepping on certain tiles at the entryway opens each temple’s doors via some sort of hydraulic mechanism. Yet they do not know why these structures are here, what they are for, or why they are the first to discover them in regions that have been thoroughly explored before.
Finally, each puzzle thus far has used a different pairing of gateway state changes and tile jumping options (see above). There are only a finite number of these, so I’ll necessarily need to start repeating, but some of the possible combinations keep the gateways open more often and some keep them closed more often. I’ve tried to stick to combinations that strike a balance. Regardless, it has been an interesting aspect of the design challenge to see how these combinations affect movement within the puzzle space.
The idea of using gerrymandering as the basis for a puzzle was not originally mine. The credit goes to my Imagineeditor at the time, Carol Blackburn, who suggested it while discussing over e-mail the topics for the twelfth volume of the magazine (2004-2005). As soon as I read her idea, it immediately struck me as a great idea for a puzzle, so much so that I checked around first to see if anyone else had made a puzzle based on gerrymandering. To my surprise, no one had1.
This is a good opportunity to discuss puzzle suggestions and what I look for in a new puzzle concept. Carol’s idea for a gerrymandering puzzle was actually the first useful suggestion I’d ever received to make a new kind of puzzle. Over the years, I’ve received some feedback and a few fan letters from Knossos Games puzzle solvers. Suggestions for puzzles tend to go along the lines of, “Hey, you should do a puzzle about dinosaurs!” While I genuinely appreciate the feedback people give me, recommendations like these aren’t very helpful, since it’s not usually obvious to me what about (in this case) dinosaurs would make for a good puzzle. My response to these suggestions is always: “No, you should create a puzzle about dinosaurs, since you clearly have an idea of what a good puzzle about dinosaurs would look like.”
So what was different about Carol’s suggestion? To answer this question, I need to explain what I am looking for in a new puzzle concept. Puzzles are always based on rules. The rules could be simple or complicated, few or many, obvious or obscure. Sometimes the stated rules for the puzzle conceal an entirely different set of rules that you can only discover by wading into the puzzle itself. Some of the hardest puzzles are those where solving the puzzle requires discerning the unstated rules by which the puzzle operates. Regardless, puzzles need rules, and coming up with new puzzles 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.
This is what struck me about Carol’s suggestion of gerrymandering. All I knew about redistricting at the time was that it had to do with the process of forming political districts, and that it could be used nefariously2. But I immediately recognized that gerrymandering must have some underlying mathematical principles that could be utilized to make a puzzle. (In fact, actual gerrymandering could be considered a legitimate puzzle in and of itself. How no one else has thought to make a puzzle out of this is beyond me.)
The first step in exploring if my instinct was correct, that this would make for a good puzzle, was to learn more about gerrymandering. I found a lot of information concerning the history of gerrymandering, less on the mathematics of how it can be achieved. I’ve lost the reference where I first learned of the techniques of packing and diluting, perhaps it was this portion of the wikipedia entry. These mathematical techniques form the basis of how political redistricting can be used to affect election results, and as such they would need to form the basis of the puzzle.
The details surrounding the actual puzzles, like using square precincts (at first called “sub-districts”) in a square territory, and having yes and no votes (rather than democrat and republican, for example), were chosen to keep the rules simple, emphasize the mathematics of redistricting over the politics of it, and focus the difficulty of the puzzles on forming the districts themselves. I also recognized early on that the instructions would need to guide puzzle-solvers through the two main techniques they would be using to influence the vote, and thus example puzzles demonstrating those techniques would be necessary.
The final secret to these puzzles is that, once the gerrymandering cover story is wiped away, all that is left are pentomino tilling problems. The 5×5 grids can be divided up in many ways, and the precinct vote totals are what influences how one decides between multiple possibilities. So while I want solvers to learn about gerrymandering, on another level, there is something to be learned about tiling problems here as well.
Imagine has run two election issues, one in 2004 and another in 2008. When I made the first gerrymandering puzzle, I actually made several of them. Coming up with a cogent set of rules for myself as to how to design this type of gerrymandering puzzle, in which there is only a single solution, was very difficult. So in most cases, when I succeed in forming a new type of puzzle like this, I make more than one puzzle. I had hoped that Imagine would run another election issue in 2012 (and I’m lobbying for one in 2016), because I have several of these to share, in both the 5×5 size and a harder 7×7 size (where districts are septominos, containing seven precincts).
During the summer of 2007, I was moving from Madison, WI, where I had just completed my graduate studies, to Dayton, OH, where I would be starting my first college professor job. To this day, I still remember driving down I-65 through northwest Indiana and seeing for the first time, with my own eyes, a wind farm. Rows and rows and rows of wind turbines stretching for as far as you could see. Even going 70 mph, it still takes a solid 10 minutes to drive through the Meadow Lake Wind Farm, which has an operational capacity of 500 Megawatts.
Once it had occurred to me to use a wind farm as the basis of an environmental puzzle, I needed to figure out how such a puzzle would really work. The details of the puzzle, like the T-shaped footprint and “windier areas”, are loosely based on how wind farms are actually built. In reality, the windiness across wind farm areas is generally quite consistent at any one time, with altitude and access to the power grid more critical to wind turbine placement.
The solution strategy for these puzzles is based on an incomplete tiling of the T-shaped turbine footprints. You can tesselate these shapes to maximize the number of turbines; however, that regular tessellation (or complete tiling) is disrupted by the irregular footprint meeting the straight boundary of the land. Plus the windier areas give some incentive to shift the pattern around, or break it up entirely, so long as you do not lose too many turbines in the process. Many of the earlierpuzzles I wrote maximized the use of the windier areas, so that every turbine could be placed (in only one way) on those areas. Later puzzles shifted away from this line of thinking and towards using the interlocking tessellation itself as a constraint to create a maximum number of turbines that could be placed in windier areas.
This puzzle is based on the television show Survivor, in case you hadn’t already guessed. During the first season, I was completely hooked. There were many things about the show I found to be deeply intriguing: the simplicity and naiveté in the first iteration, the questions of how the editing of footage and the scripting of challenges affected the “reality” of the situation, and the basic morality play of the different characters and how they interacted. It was a completely contrived yet ultimately legitimate social experiment. Many of these issues have been thoroughly explored (and exploited) in the subsequent versions of the show and the many, many reality television shows that have followed.
The format of 16 people voted off one at a time seemed to be a perfect basis for a puzzle, but it wasn’t initially clear how to construct a puzzle around that idea. I didn’t set out to write another “truth tellers and liars” type puzzle, since those were thoroughly explored, and in my opinion perfected, by mathematician Raymond Smullyan in What Is the Name of This Book? and the follow-up The Lady or the Tiger? If you like these types of puzzles, you definitely need to get those books. Ultimately, I thought the large number of participants involved plus the mix of people who sometimes tell the truth and sometimes lie (which Smullyan refers to as normals) added something unique to the genre.
I wrote the puzzle during the summer of 2000, about four episodes into that first season of Survivor. Because of publishing deadlines, I had to write the puzzle before I knew the outcome of the show, even though it would be printed after the season had concluded. This turned into something of a moral dilemma. How should I characterize the winner? Should they be honest or deceitful? What will it take to win? This all pertained to how I balanced out the true and false statements. I finally decided that the puzzle would work on the assumption that the further a person got in the game, the more likely they would be to lie, not necessarily to the other contestants, but to the media or the viewer at home (to whom the clues have been given). In retrospect, my prediction seems pretty good.
I considered many ways in which to construct the clues, such as basing them on gender and the two competing tribes. The gender clues were in the puzzle until the final revision, at which point I found them to be unnecessary. I decided against using information about who belonged to which tribe, or mergers or immunity challenge results, because I wasn’t certain how the real game would work after the tribes merged. When I wrote the puzzle, I wasn’t sure when or even if a merger would occur.
Coming up with names for people in logic problems is always a challenge, and in this case I needed sixteen of them (that had to be different from those on the show). Since I run into this problem quite often, whenever I see an interesting name, I make a mental note of it for the future. After submitting the puzzle, Carol Blackburn (my editor at the time) zeroed in on the conspicuous “Eutaw” and correctly suspected that the names were all lifted from streets (or abbreviated versions thereof) in downtown Baltimore, the home office of Imagine magazine. I had been there the previous summer and saw the sign for Eutaw Street, which borders Camden Yards, while I was at a Baltimore Orioles baseball game. When it came time to turn the generic placeholder names used in creating the puzzle into actual names, I grabbed the map from my trip and started listing off street names. There were more male names than female names, thus Holliday, Chase, Madison, and Eutaw all became women. It also meant I had to keep checking that I was using the correct pronouns, because I often forgot which gender went with which name.
Writing the detailed solution was excruciating because I kept mixing up the tenses. You see, just like on the show, these people are saying these statements in the present, but are referring to things which happened in the past. So I have to refer to the people saying their statements in the present tense, but anything said in the statements about who’s kicking off whom has to be in the past tense. Things became even more tangled when I had to refer to previous steps in the solution method, which were before the present, but after the contest. Confused? I sure was.
In editing the solution for the latest revision of the website, I discovered a more efficient solution, which made some of the clues redundant. This new version is the one currently posted to the website. More details about the rationale for the change and the eliminated clues can be found here.
You’ll also note that this puzzle is titled “Castaway, Part 1”. Part 2 was written ten years later for the Philosophy issue (v17.n4, March/April 2010), when I again felt it would be appropriate to revisit the ethical and moral issues brought up on Survivor. That puzzle will be featured and discussed in the next set of updates for the Logic puzzles.