Tag Archives: puzzlemaking

Imagine 24.4 puzzle…

Work has kept me very busy this spring, so I’m still working on updates for the If Then City Blocks and Cell Wall Transport System puzzles. But it was long overdue to switch off the winter theme and remove the direct link to the holiday puzzles, although they can still be found here.

Origin Stories: Christmas Trees

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.

Origin Stories: Reindeer Games

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.

Top: L-shaped example from two-year garden logic problem.  Bottom: L-shaped example from reindeer games holiday logic problem.

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.

Puzzle Books

The sidebar of the logic article I wrote in Imagine contains several books about puzzles. Here is an extended, annotated version of that list. If you’re thinking of buying one of these books, please consider clicking the Amazon affiliate links below:

Books by and about Lewis Carroll (Charles Lutwidge Dodgson, 1832-1898):

Two books detailing his mathematical puzzles, including sections on “game of logic”:

Lewis Carroll in Numberland: His Fantastical Mathematical Logical Life by Robin Wilson (W. W. Norton & Co., 2010)

The Universe in a Handkerchief, Lewis Carroll’s Mathematical Recreations, Games, Puzzles, and Word Plays by Martin Gardner (Copernicus, 1998)

Lewis Carroll’s two major works of logic are now published as one volume:

Symbolic Logic and the Game of Logic by Lewis Carroll (Dover Publications, 1958)

A collection of Lewis Carroll’s puzzles that remained unfinished upon his death:

Rediscovered Lewis Carroll Puzzles, Newly Compiled and Edited by Edward Wakeling (Dover Publications, 1996)

A new edition of the absolutely authoritative and exhaustive guide to both classics: Alice’s Adventures in Wonderland and Through the Looking Glass. Extensive annotations concerning the background and influences of the work, the historical context, and how the works comment on the state of mathematics:

The Annotated Alice: 150th Anniversary Deluxe Edition by Lewis Carroll, Introduction and Notes by Martin Gardner, Original Illustrations by John Tenniel (W. W. Norton & Company, 2015)

A selection of books by Raymond M. Smullyan (ordered chronologically):

What Is the Name of This Book? The Riddle of Dracula and Other Logical Puzzles (Dover Publications, 1978)

The Lady or the Tiger? and Other Logic Puzzles (Random House, 1982)

Satan, Cantor and Infinity: Mind-Boggling Puzzles (Dover Publications, 1992)

The Riddle of Scheherazade and Other Amazing Puzzles (Harcourt, 1997)

King Arthur in Search of His Dog and Other Curious Puzzles (Dover Publications, 2010)

The Gödelian Puzzle Book: Puzzles, Paradoxes and Proofs (Dover Publications, 2013)

Books about Sudoku and other number logic puzzles:

A mathematical exploration of Sudoku, including how many Sudoku puzzles there are, how many clues are necessary, and different fields of mathematics that can help us better understand Sudoku:

Taking Sudoku Seriously: The Math Behind the World’s Most Popular Pencil Puzzle by Jason Rosenhouse and Laura Taalman (Oxford University Press, 2009)

For more background on Latin Squares and other mathematical concepts that Sudoku are based upon:

Before Sudoku: The World of Magic Squares by Seymour S. Block and Santiago A. Tavares (Oxford University Press, 2009)

For a thorough guide of basic and advanced solving strategies (this excellent book is unfortunately out of print):

Teach Yourself Advanced Sudoku and Kakuro by Nick Afka Thomas (McGraw-Hill, 2006)

For many, many more examples of different types of number puzzles (another excellent book that is currently out of print):

Japanese Number Puzzles by Anthony Immanuvel (Running Press, 2006)

Masters of Deduction

Imagine magazine has graciously allowed me to share with everyone the logic article I wrote for their most recent issue!

Click on the above thumbnails to open jpegs of each page, or use this link for a PDF

I mentioned some of my own puzzles in the article:

• I’ve written several logic puzzles (organized both chronologically, in the order they were published, and thematically), including a few special holiday logic puzzles.

• 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 rubber stamp puzzles were directly influenced by battleship puzzles I had seen in Games magazine. Another example of a grid-based number puzzle, albeit one I created and not based on one of the Nikoli puzzles, are the gerrymandering puzzles. Another example of a puzzle I created that was inspired by a different puzzle are the space pods puzzles. I wrote a blog entry about how I tried to improve upon another Games magazine puzzle I had seen.

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.

Welcome Logic Fans

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:

There are “origin story” blog entries for some of the logic puzzles I’ve written: the Coin Box problem, the Two-Year Garden problem, and the Castaway (Part 1) problem. I’m working on more.

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.

Crossword Plagiarism Update

FiveThirtyEight continues to follow the unfolding story I summarized in an early blog post concerning a crossword puzzle plagiarism scandal.

On April 29, they reported that Universal Uclick, the syndicator of the puzzles, confirmed the allegations that the crossword puzzle themes had been copied from those found in the New York Times.

But the editor of the crosswords, Timothy Parker, would only be placed on a three-month leave, which led to an uproar in the puzzlemaking community.

Then on May 10, FiveThirtyEight reported that Parker was out as USA Today’s crossword puzzle editor, although they would retain Universal Uclick’s services, albeit with a new puzzle editor.


Origin Stories: DNA Transposition

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.

Origin Stories: Pathogen

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.

Origin Stories: Greek Temple

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.

GT Published Keys

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.