Roderick Kimball wrote to tell me about an interesting logic puzzle he’s developed called Path Puzzles.
In these puzzles, you must make a path that winds its way from one opening to another on the edge of the grid. The clue numbers tell you how many squares in each row or column are occupied by the path.
The more advanced versions of the puzzles involve more ambiguous clues, and multiple door choices; which makes them significantly more challenging.
If you’d like to try these puzzles out, check out this sample page that Roderick has provided to Krazydad readers:
Want some more? Go to pathpuzzles.com.
And happy puzzling!
The illustration shows 5 different sets of photos (randomly selected from Flickr) accumulating over successive rows. The first row is a single image. Then 2 images, 5 images, 25 images, and 100 images in the bottom row. I have cranked up the saturation to reveal the orange shift (unprocessed averages tend to look like dirt or milk-chocolate).
I stumbled across this effect in 2006, playing with Flickr, and have blogged about it a few times. Other digital artists who use the same averaging technique have also observed the effect. The reasons why it happens are not yet entirely clear, but I suspect it has something to do with chemistry and physics. Interestingly the same effect occurs with collections of human-generated synthetic, abstract art (not photos), such as fractals.
Over thanksgiving break I wrote an informal paper describing my findings to date, in hopes of attracting some brighter minds to the topic. Perhaps you are one of those minds? You’ll find my paper here.
If you prefer pretty pictures to words, here’s a Flickr set containing some of my image averaging experiments.
PD Weekend is an upcoming series of events at CRASHspace in Culver City revolving around electronic music and art made using PureData, it’s commercial sibling MAX/MSP and other tools. The creator of both of the aforementioned applications, Miller Puckette will be on hand to teach a Raspberry PI + PD seminar. There will be a number of other great classes covering topics such as Live Sampling, Laptop Orchestration, and Creativity.
The events run from November 13th, through 17th, with most of the events on Saturday and Sunday.
Saturday evening there will be a lovely concert of Handmade Music, and on Sunday afternoon there will be the monthly PD Patching Circle – a great free event where you can hang out with other handmade music enthusiasts, and work on your projects.
At the Saturday evening concert, I’ll be demonstrating a new-improved version of my astronomical-music-clock Wheel of Stars (I’m tarting it up right now…).
Hope to see you there!
An interesting puzzle, basically a kind of maze with limited movements, that resembles a crossword, by Tyler Capp.
John Welch writes an excellent blog for advanced Sudoku Solvers called Systematic Sudoku.
Every Tuesday he posts a new detailed analysis of a tough puzzle, one of his goals being to understand the comparative difficulties of various sudoku publications. Sudoku difficulty is hard to measure and describe, and different publications (including this website) use different metrics and language to describe puzzle difficulty. How do the “Super Tough” puzzles that I offer here compare to the “Diabolical” puzzles you might find in your local newspaper? John Welch aims to find out.
Today, I had the honor of seeing John review a puzzle from this website: Namely Super Tough Volume 5, Book 1, Puzzle 5. If you’re new to John’s website, you will probably find his notation system, reminiscent of algebraic chess notation, hard to understand. He explains the notation elsewhere on his blog. I suggest reading the introductory links he has posted at the top of his blog, as well as some of his earlier posts on easier puzzles, to help get a feel for his lingo.
Today’s review is the first of a series, and you’ll see a few follow up reviews of Krazydad puzzles in the coming weeks. I, for one, am looking forward to see what John has in store for the next few Tuesdays!
Link: Systematic Sudoku
It’s been a long while since I’ve posted, but hopefully, there are still one or two of you out there who understand the value of a good RSS feed. I’m back to write about a book I’m enjoying, with the following cryptic title:
If this line of BASIC code already means something to you, then you are probably going to enjoy the book, go buy a copy right now, the proceeds go to a great charity. If the title looks like a bizarre and useless incantation (it is!), read on.
The book is an exploration of its title: A one-line BASIC program that was intended to run on an early 80s Commodore 64 computer. The program produces the maze-like pattern shown on the book’s cover, and its inner sleeves.
Each chapter explores a different facet of this program, and by doing so it covers an incredible amount of ground. There is a chapter on mazes, a chapter on randomness, a chapter on grids, a chapter on the BASIC language, and so on. If you think this is a lot of pages to devote to a one-line computer program, you are mistaken. This is not a long book, and it barely scratches the surface of each of the diverse subjects it touches upon, from Falcon looms and Truchet tiles, to 8-bit computers and flying toasters. There have been many books written about mazes, and whole careers built upon the study of randomness. Here each of these subject gets just a little chapter.
It is the surprising depth and far-reaching ramifications of little useless programs like 10 PRINT that got me into this game, back in the early 80s. I had a Timex Sinclair, and a VIC 20, both purchased at K-Mart, and I fondly remember writing one-liners like these, staring into the glowing phosphors of a little television, until I could barely keep my eyes open in the early morning light. During the months that I manipulated those phosphors, the symbols they represented were manipulating me. My fevered brain underwent more intellectual growth during that period than any time in my life since my early childhood.
The book was written by a team of what my colleagues call “unicorns” – cross-disciplinary people who bridge the worlds of creativity and technology. I was expecting a set of disconnected essays from different voices, but I didn’t get it. The authors used a Wiki to collaborate, and the book feels as if it were written by a single, extremely erudite, polymath author. The chapters cover separate subjects, but the whole is very much connected, helped by it’s extremely constrained subject – that single one line program. Although the book necessarily describes some technical subjects, it is written for a lay audience.
I think of myself as a unicorn. There are a lot of us out there, but we are not as common as I would like. My feeling is that unicorns provide an important bridge between the humanities and the sciences, and that unicorn skills should be nurtured. All of my professional career, I have obsessed over a set of subjects which were, until recently, not given sufficient attention in the computer science press.
For example, I’ve always been fascinated by the RND() function in the BASIC language – I initially thought it was the most important feature of the language. For a long time, the amount of joy I derived from writing software was proportional to the amount that the features depended on randomness. There is a relationship between the RND() function and the perception of utility. To me, programs that are useful, and that do not require randomness, are useful, but boring — they fill out your tax return and monitor patients in hospitals. The RND() function is like a firehose from God, and the programs that use it are useless, but fun — they are games, and simulations, and art.
So, as an auto-didact (as many unicorns are), I was surprised that in programming texts that describe programming languages, the RND() (or rand() or random()) feature is always given such brief, almost dismissive treatment. I’ve even met programmers who (gasp!) have never used it! Meanwhile outside of programming language texts, the topic is barely discussed. It’s not a topic that non-programmers have been exposed to. To me, it’s the first thing you should learn as a neophyte programmer — it opens up programming in a big way. Yet so many computer science students are not exposed to it early enough – instead, they are compelled to write functions which factor numbers and store fractions, and do other numeric manipulations. Many programmers have thrived in this sterile environment, but it doesn’t suit unicorns. For us, making random poetry and assembling Truchet tiles is a much better follow up to “Hello, World.”
The only way to really appreciate the wonder of what you can do with RND() is to write some code. Immerse yourself in it. Play with it. It’s programming as play, rather than programming as work. And it’s more fun to play with something when it’s easy — when there are virtually no barriers to entry. The one line BASIC program was my E-ticket to this world of wonder. Useless one line BASIC programs are a kind of activity that used to be called “recreational programming”. In addition to the C64 magazines discussed in the book, there used to be a wonderful recreational programming column in Scientific American by A. K. Dewdney (I was a little late for Martin Gardner); half my career owes its existence to these publications, and especially Dewdney’s column – it was my principle reference.
Another feature of unicorns is that they don’t mind using programming techniques that are no longer on the list of officially approved methodologies by the software engineering orthodoxy. Incantations are only a means to an end. We are not in the business of making incantations, we are in the business of making universes, using incantations as a tool. 10 PRINT, for example, contains a GOTO command. GOTO, of course, has been the bane of readable code almost since the second edition of “The Elements of Style”, and the BASIC language itself sits on a lowly plain of derision slightly above COBOL. The book also addresses the unfortunate gulf between recreational coders and the computer science establishment.
Ultimately, it is the love of RND() that separates this particular creative coder from your dyed-in-the-wool computer science nerd. At this late stage in my creative programming career, I no longer make as much use of RND() – I’ve discovered new ways to achieve the same important thing it gave me: complex and beautiful behavior with very little effort. The holy grail of the unicorn is the perfect one-line program. The one-line program that succeeds in recreating the universe, and making its own DNA, and breeding with itself so that a new sub-universe is born. This is our philosopher’s stone.
What the book helped me to remember is that joyful process of trying to create the universe with a line of code. Each time, I would get something interesting, like a labyrinth or a kaleidoscope; but I wouldn’t quite get that ultimate spark. So I would try again, never quite thinking that I had succeeded in creating the universe I sought. The book shows that the universe was already there — like the grains in a Mandelbrot filament, each of those little programs already contains a little universe, you just need to pay attention. Zoom in the right direction, and you’ll see it.
Unfortunately, one line programs, like unicorns, have become an endangered species. Those of us who remember one line BASIC despair at the new hurdles that have been raised, which prevent young people from discovering the joys of the random number generator. When we expunged GOTOs from the reserved words of all the new programming languages, when we made our code structured, object-oriented and useable for large complex software engineering projects, we also made it much harder for kids and teens to use those same technologies to explore the imaginary landscape. If the first programming language I had been exposed to was Java, I think I might have ended up in a different profession entirely.
The proceeds of this book go to PLAYPOWER – a charity which aims to give disadvantaged kids access to extremely cheap computers that have a one line BASIC. This seems like a wonderful thing. Damn, I want one of those computers too!
I miss my VIC 20.
I recently added a new puzzle variety to Krazydad, called Jigsaw Sudokus. These are very similar to normal Sudokus. Like those puzzles, each row and column contain all the digits 1 through 9. However, these puzzles are missing the 3×3 blocks found in regular Sudoku. Instead, the puzzles are divided into a set of irregular “jigsaw” pieces – each of which also contains the digits 1 through 9.
In my collection, the Jigsaw layouts are always different, so each puzzle offers a unique challenge which you won’t find in regular Sudoku.
Because these puzzles are a little different, I thought I’d offer a short tutorial to help you get started. I’ll start by solving just a couple squares of Puzzle #3 from book 1 of the Tough puzzles.
In particular, I want to demonstrate the one unique strategy that these puzzles require, apart from the regular Sudoku strategy you already know. This is called the “Law of Leftovers”, or the “Rule of Overlaps”. Take a look at the puzzle, below.
You’ll see that I’m interested in the jigsaw piece on the lower left, which I’ve colored mostly in blue. When you compare this piece to the first column, you’ll see that 3 cells stick out on the bottom (they contain a 3, a 6, and blank). I call these cells “outies”. There are also three cells left over at the top of the column (“innies”), which contain two blanks, and an 8. I’ve colored both sets of cells pink.
What may, or may not be obvious to you, is that the innies and the outies must contain the same set of 3 numbers. Why? The jigsaw piece must contain the numbers 1-9. The first column must also contain the numbers 1-9. The 3 outies must contain all the numbers from 1-9 which don’t appear in blue, and the 3 innies must also contain the numbers from 1-9 which don’t appear in blue. These three numbers must be the same, in both cases.
Since the innies contain an 8, the outies must also contain an 8. Since the outies have only one blank, and no other 8s, the blank on the bottom must be the 8. Similarly, the two blanks in the innies must contain, collectively, 3 and 6.
This takes us to step 2 (next picture).
Here we have another case of innies and outies. In this case it involves two jigsaw pieces, which overlap with the two columns on the right. The Law of Leftovers can work with any number of puzzle pieces, and any number of rows/columns, as long as the number of pieces, and the number of columns are the same. Once again, the outies on the left, in purple, must be identical to the innies, in pink.
This means the one blank cell in the outies, must be 2, and the two blanks in the innies must be, collectively, 3 and 4. This brings us to step 3, shown below.
As I already mentioned, you can use the Law of Leftovers with any number of rows or columns, as long as the number of pieces you are overlapping is the same. Here I am overlapping 4 puzzle pieces with the top 4 rows. You’ll see that there is one outie, in purple, and one innie, in pink.
At this point, I don’t know what those two numbers are, but I know that they must be the same. This means that any constraints that apply to one cell must also apply to the other cell. Since I know that the innie, in the first column, can’t contain the numbers 1,2,3,4,5,6 or 8, I know that the outie also can’t contain those numbers. Thus I can narrow both cells down to the choices 7 and 9.
When solving these puzzles, I sometimes find it useful to leave known identical cells colored with the same color, which I’ll do here.
I’ll end the tutorial here, and give you a chance to find more outies and innies on your own.
I *do* want to mention that lately I’ve been doing these puzzles on an iPad, using the application UPAD. There are a few advantages to this method.
1) I get as many colored pencils and colored highlighters as I like, which is useful for jigsaw strategy,
as well as other advanced strategy.
2) If I make mistakes, I can erase/undo the colored markings easily. I can switch colors and do a few “what if” markings, then hit undo until I get back to the original color.
3) I can draw on each page of my books, and it leaves the drawings on each page intact.
4) No wasted paper/ink.
5) It still feels a lot like using pencil and paper, which I love – especially when I use a stylus.
If you combine a sufficiently large number of (uncorrelated) photos, and choose the images carefully, you can approximate any image you like.
This “Shroud of Turin” was made by carefully choosing 300 photographs from a larger set of 20,000 photographs of graffiti. All images come from Flickr and are Creative Commons licensed. The images are combined by adding the individual pixel values, so that each of the 300 photos contributes an equal amount to the final image.
You’ll notice an orange color that appears a few seconds into the video. This orange color is usually seen when combining uncorrelated photos. I blogged about this phenomenon a few years ago.
I wrote the software to produce this video in Processing.