The first Greek 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 Imagine editor 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).

The first Wind Farm puzzle was created for the Spring 2008 green issue of Imagine. Similar to the Gerrymandering puzzles originally created for the election issue in 2004, I wanted to create a puzzle that would be interesting on its own, but also draw attention to an important subject. I considered several possible environmental topics at the time: carbon emissions, melting glaciers and rising sea levels, endangered species, climate change. But an indelible memory was fresh in my mind, and it turned my thinking toward my own experiences in sustainable energy.

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 earlier puzzles 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.

While the two-year garden logic puzzle was written while I was teaching at the Bethlehem, PA (Moravian College) CAA site during the summer of 1999, the idea for the puzzle came on a bus ride.

There was a lot of time to look at the passing corn fields while on my frequent Van Galder bus trips between Madison, WI and O’Hare Airport in Chicago. It made me think of the huge vegetable garden I helped my dad with every year when I was growing up. I remembered the plans he’d draw up in the winter, with paper and pencil in one hand and the seed guide in the other. The time spent planting in the spring. Fighting the weeds. Too much rain, then not enough. And finally, the big fall harvest. Those backyard experiences were the inspiration for this puzzle, where the layout of the crops had to be reconstructed from scribblings on the backs of seed packets. I went through several lists of possible crops to use, trying to find common vegetables that could be found in someone’s little garden or a large farm.

You might have noticed that the title of the puzzle is the “two-year” garden puzzle. My first attempt was a three year version that occupied twelve plots in a four by three grid. After testing it out on some of my best students that summer, I realized that puzzle was really, really tough. So I created this two-year, nine-plot version. It works basically the same, but is quite a bit easier. I still have the three-year version. It’ll turn up eventually.

Finally, those students from the summer of ’99 read the instructions and immediately changed “a farmer” to “Farmer Tim”. I do have a bit of a green thumb from all those years of helping my dad, but I didn’t keep the change since I consider myself a farmer. I kept it because they insisted!

This is an origin story within an origin story. It also required a significant amount of detective work, in more ways than one.

I was initially hesitant to write any logic problems for Knossos Games at all. There is a vast history of logic problems, with a great deal of diversity and detailed analysis of practically everything. I have high standards for the puzzles I write, especially in terms of their uniqueness and originality, and I wasn’t sure if I could add anything to this category of puzzle. A particular logic problem that I saw convinced me otherwise.

The Coin Box problem was my very first attempt at a logic problem. During my early days of puzzle making, I tended to emulate or improve puzzles I encountered, and this one was no different. I created this problem based on another logic problem, but I never had a copy of that original problem, and that’s because of the way in which I encountered the problem.

During the summer of 1997, I was a teaching assistant at the Frederick, Maryland CAA site for a class called “Mathematical Investigations”. (Sadly, this course no longer exists. This discrete math course currently offered is the closest, but there is only a small amount of overlap in the content, and I have no idea if they are taught in the same way. I should also note that CAA doesn’t exist anymore either. CAA summer courses are now called CTY: Academic Explorations, while the original CTY summer courses are now called CTY: Intensive Studies.) These summer program courses meet all day long for three weeks, so in addition to the main topic we covered each day, there were a lot of supplementary activities.

One of those was a recurring competition between the students. The class was broken up into a few teams, and the instructor and I would give them a new puzzle or math problem every three minutes on the overhead projector. The members of each team would have to work together to make sure they copied down the relevant information needed to solve each problem as well as actually solve them all before the competition ended. Of course, the team with the most correctly solved problems won.

This is where I saw the logic problem that inspired the Coin Box problem. Remember, I never got a copy of it at the time. Here are the notes I have about the problem, which were jotted down at some point well after I wrote the Coin Box problem:

One problem we used in one of the competitions had a grid of 16 numbers. The puzzle had a couple of simple clues about how to arrange the numbers in the grid. The problem was only looking for one specific number, and thus it was unnecessary to fill in the entire grid to get the answer. I wondered if I could make a similar puzzle, only make it so that each clue was necessary to complete the entire grid.

That description constituted the brief yet official history of the Coin Box problem. Years later, I found a book of problems that seemed very familiar. Baker Street Puzzles by Tom Bullimore is a collection of short problems, puzzles, and brainteasers using Sherlock Holmes and Watson as a cover story.

I immediately recognized a few of the problems in the book as ones from the competition. I dug through my filing cabinet and found a solution sheet for the problems we used in one of the competitions. The problem numbers for each solution matched those given in the book, so I knew I had found the source of the problems the instructor used.

However, absolutely nothing in the book matched the above description I had written about the source puzzle. I couldn’t remember if the competitions used problems exclusively from this book, or mixed and matched from other sources. The problem in the book that seemed closest to my description is the following:

Is this where I got the idea for the Coin Box problem? Perhaps I misremembered the details from the problem seen in competition. If this is the source problem, why would I have found inspiration in this particular puzzle? Only in trying to solve this puzzle did I start to realize why I might have wanted to recreate it.

Look at clue #3 above: SMITH’S box was also above GRAY’S (although not directly). What exactly does that mean? What is the difference between “directly above” and just “above” (not directly). Clues #1 and #6 also give a distinction between “directly to the right of” and “to the right of” (not directly).

I can think of two possibilities. In one interpretation, “directly above” could mean that the two boxes are touching, where one is right above the other, while “above” could mean in the same column, but not necessarily touching. (Saying that one box is above another does not preclude that it is, in fact, directly above it, so this doesn’t force the “above” box is in the top row and the “below” box is in the bottom row.) Alternatively, “directly above” could mean in the same column, while “above” could simply mean that the “above” box is anywhere in a row above the “below” box. These interpretations are very different! And nowhere in the puzzle is it clear which we should use.

I decided to chart out all the possibilities. Depending upon how you interpret “directly above”, “directly to the right of”, and “between”, this yields many, many solutions to the problem, shown below. What I believe to be the intended solution is highlighted in blue.

As it turns out, for the solution given (that MORIARTY’S box is #9, matching the above presumed solution), “directly” is used consistently to mean touching, although sometimes this is a choice, and other times clues force you to interpret “directly” as touching. However, the generic “to the right of” (clue #1) is forced to be in the same row, while the generic “above” (clue #3) is not in the same column. Again, there is nothing in the stated clues that makes any of this clear, and thus I would claim that there are alternative, valid solutions of boxes 5, 6, 7, 10, and 11.

In the Coin Box problem I wrote, it seems fairly clear that I was responding to this language problem. The clues I used only contained references to “directly” above and below, and never generally above and below. (I also added several clues that relied on numerical properties of the values involved.) In updating the puzzle for the new website (and in thinking through this blog post), I decided to include language defining “directly”. Even though its usage can be deduced from the puzzle itself, since in several cases “directly” must mean contiguous and thus should be used consistently, that possible ambiguity is not what I want to be part of what makes the problem difficult.

It is still possible that this problem is not what generated the Coin Box problem. I think it is more than likely, given the similarity of the clues and the correction of some of the source problem’s ambiguities. Even with the distorted narrative I had in my notes, there is more here that verifies the origin of the first logic problem I wrote than refutes it.

By the way, if we don’t enforce the interpretation of “directly” as contiguous in the Coin Box problem, then there are two alternative solutions to that as well. Clues #1, 5, and 7 could be used differently to produce these solutions (differences highlighted in blue):

New ideas for puzzles come to me from anywhere and everywhere. Often times they come from everyday experiences, and through ideas I contemplate at my actual job, but I also like to check out puzzle books, magazines, and websites to see what other puzzlemakers are doing. Sometimes I see a puzzle that’s great, and I want to emulate it. Other times I see a puzzle that was based on a promising idea but was (in my opinion) poorly executed, and I think I can do better. In this continuing series, I’ll share where my new puzzle ideas come from.

There is one more story about the Space Pods puzzles I wanted to tell before moving on to other things. I dug through some stuff and managed to find the puzzle from Games magazine that inspired the first Space Pods puzzle.

I thought this puzzle had some interesting ideas. However, I also thought the way it was constructed was convoluted and cumbersome. I like complex puzzles, but I find ones that have overly complicated rules to be tedious and unnecessarily torturous.

As you can see in the instructions above, three people move around the hexagonal grid unlocking doors. But each person has a certain number of specific keys, which can only go in specific locks, and once they run out they have to return to “base”, and then some of the doors re-lock, but some stay open. These details have to be summarized in a chart for each puzzle.

This puzzle is difficult because you have all of this junk that you have to keep track of. (The author even senses this – note that the instructions contain a suggestion of how to keep a record of which doors are unlocked.) Not exactly my idea of a fun puzzle-solving experience.

Despite this sentiment, I thought that there were some unique ideas contained in the puzzle that I wanted to explore. I had used two people helping each other in the Double Jumping puzzles, so why not try three people? I liked the hexagons, because it would open up more possibilities for connecting rooms together compared to a square grid. And I liked the idea of doors and locks that would be unlocked and re-locked.

I kept the three people, but now they work together, instead of sending one out, then another, then finally the third person. And the objective is no longer that just one person makes it to the goal (which never made sense to me), but that all three have to get there. I ditched the standard hexagonal grid by adding hallways. This helps connect non-contiguous hexagons and opens up the puzzle, but also precludes using a distance-to-goal heuristic while solving, which makes the puzzle harder. The most important change was simplifying the procedure of unlocking and re-locking doors. Instead of charts explaining who had what keys, the door locks would be dependent upon where the people were in the puzzle space. Thus the “chart” becomes part of the puzzle space itself (which makes it a lot easier to keep track of who can open which doors when). Consequently, the rules are vastly simplified, but the puzzles can actually be more complex, depending upon how large the space is and how many steps there are to solve the puzzle .

All in all, I think it’s a big improvement. My thanks to Sky Williams and Games magazine for inspiring me to make a puzzle using three people unlocking doors together. Oh, and in case you were wondering, here’s the solutions to the above puzzles.