Block by Block Jr. is a 3d construction puzzle based on the classic Soma Cube puzzle. Like the Soma Cube, Block by Block Jr is a great way to learn to visualize shapes in 3 dimensions — an important skill in science, math and engineering. For instance chemists visualize molecules, mathematicians visualize data, and architects visualize buildings and other structures. Solving Block by Block Jr. puzzles also develops problem solving and teamwork skills.
Every year my daughter’s elementary school stages a day of hands-on science exhibits called Discovery Day. For this year’s event I decided to build a giant 3d puzzle, inspired by giant Soma cubes other people have built out of cardboard boxes. However, when I tested the Soma cube on kids, I found it was much too hard. Adults too had trouble with it.
So I developed a simpler version of the Soma Cube, with fewer, smaller pieces, which I call Block by Block Jr. The original Soma cube has seven blocks that form a 3x3x3 cube. Each block is made of 4 cubes, like a Tetris block, except for one block that is made of 3 cubes. Altogether the blocks contain 27 cubes, and can be assembled to make a 3x3x3 cube.
Block by Block Jr has six blocks: three blocks made of 4 cubes each, and three more blocks made of 1, 2, and 3 cubes. Altogether the blocks contain 18 cubes, and can be assembled to make a 3x3x2 rectangular solid. Block by Block Jr puzzles are definitely easier than Block by Block puzzles, but the hardest ones are still quite challenging. Here are the six Block by Block Jr pieces.
For Discovery day I built three versions of the puzzle: a giant version featuring built out of 10″ cubical cardboard boxes stuck together with duct tape,
a half-sized version made of 5″ cardboard cubes,
and a small version made of 1″ wooden cubes
In all three versions of the puzzle I stuck colored dots onto the blocks to make it easier to identify which block is which.
Here are instructions for making your own set, including where to order cardboard boxes and wooden cubes.
And here are challenge sheets containing puzzles that require 2 to 6 blocks. Caution: there are answers for only the first 5 (of 10) challenge sheets.
You are welcome to make copies of this puzzle for personal use or for gifts, but please do not sell them…I’m working on turning this puzzle into a product.
Three years ago, the Mathematical Sciences Research Institute at UC Berkeley in California decided to do a little outreach, to share the wonders of math with the public. Little did they know that 20,000 people would show up at the Smithsonian Institute in Washington DC for the first National Math Festival. Yes, there’s that much interest.
Now, two years later, interest has only grown, with movies like Hidden Figures, and snowballing interest in STEM (Science, Technology, Engineering, Math) for education. On Saturday, Apr 22, 2017, another 20,000 people descended on the DC Convention Center (and dozens of affiliate science museums across the country) for the second National Math Festival. There were balloon polyhedra, talks by mathematicians profiled in the movie Hidden Figures, mathematical magicians, and mathematical art from all over the world. It’s all the fun stuff about math that you never learned in school.
I hosted an all-day hands-on workshop showing kids of all ages (that includes adults) how to make your own versions of Sudoku and Pentominoes — two of the most popular mathematical puzzles. If you didn’t make it to the event, or you attended and want more, here are the downloadable handoutsso you can try them at home, plus the twelve 11″x17″ play mats, and instructions for printing the mats and making the accompanying manipulatives. If you want more, check out the links at the end of this article to math puzzle books and web sites.
Puzzles are the literature of mathematics. Puzzles are where the ideas in math class come alive, and dance off the page into our imagination. Traditional math education is full of worksheets that are boring to fill out (word problems are third-rate puzzles), and train kids in very narrow ways of thinking. Puzzle, in contrast, are exciting to solve, memorable, and require resourceful think-on-your-feet problem solving.
Puzzles are not just entertaining, they have a deeper purpose. As every scientist, engineer and mathematician knows, puzzles are the play version of problem solving. When kids solve puzzles, they wrestle with the elements of problem solving that they will encounter later in life, just as lion cubs wrestle with each other to build fighting skills they will use later in life. And problem solving is one of the most important skills that every child needs to succeed in life.
Playing puzzles and games is as important for a well-rounded math education as reading books is to language education. In an ideal world, math students would spend much time playing puzzles and games, doing art projects, building things, and reading math stories.
If that sounds strange, imagine an English class without books. If English class consisted entirely of spelling, grammar and verb conjugation, then students would become proficient at the mechanics of language, but never fall in love with the subject. They would ask “when are we ever going to use this?” And they would not be able to read anything but sentences written in the forms that appear on their worksheets. Language education without books sounds absurd. But sadly, that is exactly how conventional math education works.
WHY PUZZLE MAKING?
If solving puzzles is like reading, then, making puzzles is like writing. Making a puzzle is harder than solving a puzzle, because you don’t have the safety net of knowing what the solution is, or even if there is a solution. It’s like a writer facing a blank page, or an artist facing a blank canvas. The challenge is higher, but so are the rewards.
Making your first puzzle is an important experience that everyone should have — it changes mathematics from a utilitarian chore into a creative activity that you can make your own. Making a puzzle is like writing a story or composing a song. Doing it well takes years of practice, but anyone can try it if they start with small steps. In my puzzle making handout, I start simple — make a 4 by 4 Sudoku puzzle, or a 5-piece Pentomino puzzle. For those who want to go deeper, I offer suggestions for customizing your puzzle or changing the rules.
Having taught dozens of puzzle making classes to kids, I have found that I don’t have to do much instruction — once I tell kids that we are going to make puzzles, most kids know immediately what they want to do. Some kids focus on art and story, creating a narrative frame around a familiar puzzle. Other kids go deep into the math, making difficult puzzles with original rules. Besides Sudoku and geometric puzzles like Pentominoes, logic puzzles are particularly popular among young puzzle makers, because they are easy to construct, and easy to adapt to a particular theme.
I invite you to try making your own puzzles, and find your puzzle making style. Be sure to have someone else try your puzzle, to make sure it is solvable and that the instructions are clear. As with creative writing, your first draft is rarely your final draft. Send me photos of your creations and I’ll add them to this site. Happy puzzle making!
WHAT MORE CAN I DO?
Kids. Play lots of different kinds of puzzles. Play with friends. Find out which kinds you like.
Parents. Read math stories and play puzzles and games with your kids. See this bibliography for suggestions.
Teachers. Have a puzzle corner in your classroom. Start the day with puzzles. Don’t just give puzzles to advanced students to do when they are done with their work; puzzles are for everyone. Have kids invent test questions — the best way to learn is to teach. Here are other ideas for using puzzles in your classroom.
Quintillions, Gamepuzzles.com. Not cheap. A sublimely pleasurable laser-cut set of wooden Pentominoes, lovingly designed by Kate Jones, whose site gamepuzzles is a treasure trove of artistically crafted mathematical puzzles.
PUZZLES & RECREATIONAL MATH
BedtimeMath.org. Brief daily stories and questions for elementary-aged kids and their parents. Web site and app.
The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems. By Martin Gardner, W. W. Norton, 2001. The ultimate recreational math book. For high schoolers and adults.
ThinkFun.com. Makers of Rush Hour and other superbly entertaining physical math puzzles for kids and adults.
Mathfair.com. How to stage a math fair in your school where every child makes and hosts a puzzle.
Puzzlecraft: The Ultimate Guide on How to Construct Every Kind of Puzzle, by Mike Selinker and Thomas Snyder. A superb practical book with detailed advice on how to create nearly every type of paper and pencil puzzle, from crosswords and Sudoku, to logic puzzles and word puzzles. The only book of its kind.
In the book and movie The Martian, astronaut Mark Watney is stranded on Mars when the rest of his crew barely escapes a sandstorm in which Watney is injured, and left for dead. Surviving in a hostile alien environment takes every ounce of courage, ingenuity and humor he can muster, invoking biology, chemistry, physics, and duct tape. The author Andy Weir, the scriptwriter Drew Goddard, and director Ridley Scott do a magnificent of avoiding (in Andy’s words) “sounding like a Wikipedia article”, and instead create a rousing (in Drew’s words) “love letter to science”.
After seeing the film, it dawned on me that the entire film is a glorious example of how to teach mathematical problem solving through story telling — something I believe is much needed in mathematics education. And indeed I found several articles online discussing theMath of The Martian, andThe Martian is full of Word Problems. Here are five lessons all math teachers can learn from The Martian. (Warning: Spoilers ahead.)
Make the viewer care about the problem. Word problems in math class are lousy stories — mere equations with names attached — so students race through them without bothering to think carefully. In the Martian, you care about the problems Watney faces because you want him to survive, and because you are intrigued by the unfamiliar other-worldly setting. The classic educational game Oregon Trail uses a similar plot device — you are a pioneer leading a team of settlers out to the American West — to draw students into a singularly memorable educational experience.
A good way to make students care about word problems is to put student names and interests into the problem. It’s a simple trick, but it helps. The app and web site Bedtime Math captures student interest by leading with a popular image or intriguing news story, then asking questions about the story. You can do something similar by basing word problems on current events, or topics you are studying in class. You can also have students make up problems for others to solve — students are more motivated to solve problems that came from other students.
Let the problems lead. Problem solving is naturally dramatic. You encounter a problem you don’t know how to solve. You try something. It doesn’t work. You try something else. Eventually you have an insight and triumph. That’s the dramatic arc of story telling, and also of problem solving.
Author Andy Weir constructed The Martian by solving the unfolding sequence of problems that he put in front of his protagonist. Watney solves his food problem by farming potatoes. But where will he get water on Mars? Andy had to call on his rusty knowledge of chemistry to solve that one, and early readers corrected his mistakes. He let the actual science guide the plot, which led to a delightfully plausible story.
Just as Andy did not in advance where his plot was going, you as a teacher do not need to know in advance all the answers to the problems you pose. Instead, show your students that it is okay to struggle, to not know, and follow the problem wherever it leads.
Make thoughts visible. Every problem Watney solves has visible consequences. Plants grow. Warmed vehicle interiors allow him to wear less clothes. Rocket fuel explodes when the chemical mixture isn’t quite right. In contrast, solving a math problem on a worksheet has no visible consequence, so it is hard to care about the result. In Dan Meyer’s TED talk Math Class Needs a Makeover, he shows how to turn a drab word problem into an arresting visual experience.
Show the emotion. The Martian includes plenty of technical detail without ever losing narrative momentum. How? First, there is plenty of drama in the numbers. To survive Watney needs 1,500 calories a day. Growing potatoes will provide him with 115,500 calories. That will extend his food supply 76 days, which alas is far short of the thousand or so days he needs to last.
But more importantly, Watney is an astronaut, a human with a preternatural ability to focus on solving problems as they come up, even in the face of death. Especially in the face of death. So his palpable enthusiasm for numbers is an expression of his character. And of course what we’re really seeing here is author Andy Weir’s infectious zest for problem solving. By showing us the full range of Watney’s emotions, from joy to despair, the author keeps us engaged in the story.
Many teachers are self-conscious about showing the full emotion of problem solving, because that means admitting that you don’t always know what to do, can be bored or intimidated by problems, and sometimes make mistakes. But if we want students to be able to feel comfortable with uncertainty we need to start by modeling it ourselves. If we always act like we know exactly what we are doing, then students will do the same, covering up their confusion by pretending they understand, and never correcting their own misunderstandings.
A good way to make everyone feel safe with expressing emotion is to present puzzles that you have not tried yourself, and that require original thinking. That puts you and the class in the same position. Your role as teacher is to coach students to ask good questions. And you can do this confidently even if you don’t know the answers yourself.
Show the messiness. This is where The Martian really shines. Drama is about things going wrong. And in The Martian just about every type of thing that can go wrong does: from faulty weather prediction, freak accidents and mistaken assumptions (calculating the oxygen in the habitat without accounting for his own breathing), to political maneuvering (cooperating with China), logistical tradeoffs (shaving days off a production schedule), and ethical decisions (save Watney or save the other five crew members?).
And this is where schools get it all wrong. Traditional math classes prepare students to answer highly manicured, abstract problems like “find solutions to the equation x^2 – 5x + 6 = 0”. To this end, worksheets remove all messiness: no story, no context, no multiple solutions, only problems that fit a specific format. The result is students who memorize formulas without understanding what they mean, and cannot apply their learning to situations that differ even slightly from the forms they see in textbooks. By protecting students from the messiness of problem solving, we cripple them, and give them the false impression that problem solving is neat and efficient.
For students to master mathematics, they need to be experience math in the wild — full-bodied situations with real consequences and no neatly prepared roadmaps. Only by encountering meaningful problems, failing, and recovering, can students learn what it means to survive in the world mathematically. In the future I hope to see more stories like The Martian woven into mathematics education.
On Feb 12 I posted a puzzle canon — a single melody that can be copied so it harmonizes with itself. For instance, this famous portrait of J. S. Bach shows him holding a puzzle canon.
Here is the full story of Bach’s canon, which is closely related to a group of canons Bach wrote on the bass line of the Goldberg variations.
I based my canon on the theme of The Musical Offering by J. S. Bach. Here it is in puzzle form.
Shown below is the solution. This canon has three voices — the second and third voices are copies of the first voice, progressively delayed by 6 quarter notes, and moved down an octave. In the first 32 bars of the written-out score the three voices enter one at a time. In the second 32 bars all the voices are sounding in an infinite loop. I added a brief closing cadence in the recording to end the canon. My goal in writing this canon was to try something Bach did not cover in The Musical Offering; in this case the new twist is to employ three canonic voices instead of just two. In case you are wondering, the canon does not work if you add a fourth voice.
Shown below is the first voice of a new three-voice canon, based on the theme of Bach’s work The Musical Offering. Your challenge is to figure out how to transform the first voice to make the second and third voices. To transform a voice you may delay when it starts by some number of bars, transpose it up or down by some interval, invert the intervals so the melody goes up where the original went down, reverse it in time (retrograde), or change the speed (augmentation or diminution, usually by a factor of two). The three voices, played simultaneously will make a harmonious whole, as they do in familiar canons like Frère Jacques. For a full introduction to canons, see Norman McLaren’s wonderful animation Canon or my talk for the Museum of Mathematics about symmetry in music and art. Please email me if you find the solution; I’m curious how hard it is to solve. I will post the solution on Bach’s birthday, March 31. (For those of you who think Bach’s birthday is on March 21, you are correct, but scholars analyzing how the calendar system changed between then and now have calculated that his birthday actually fell on what we would now call March 31.)
Musical Offering Canon 1: by augmentation
I have now written three canons on the theme of the Musical Offering. Here is the story of the first Musical Offering canon.
In 1975 I had the pleasure of meeting Doug Hofstadter at Stanford University. I was an undergraduate enamored of Escher’s art, Bach’s music, and Gödel’s mathematics (I had read Nagel and Newman’s book Gödel’s Proof the previous summer, in Hawaii). The encounter was an uncanny meeting of minds; as Doug recalls we were a bit suspicious of each other, having encountered wacky people with similarly eccentric interests.
But the connection was real and we became fast friends. I spent many happy evenings working on my homework at a computer terminal alongside Doug, who was writing his magnum opus Gödel, Escher, Bach. To help him write his book he employed a classic professor trick: teach a course in order to motivate you to write the book. Each week new dialogs and chapters would appear hot off the dot matrix printer.
A couple years later he taught the course again, and I helped out as a teaching assistant. We decided to open the class with a re-enactment of the fateful meeting of Bach and King Frederick the Great of Prussia, in which the king sat down at a new-fangled pianoforte and challenged “Old” Bach to improvise a fugue on a long chromatic theme he had composed. Bach not only improvised a 3-voice fugue on the spot; three months later he sent the king an incredible manuscript called The Musical Offering, which included a six-voice fugue, a trio sonata, and ten canons, all on the royal theme. The full story is told in the dramatic the book Evening in the Palace of Reason. Here is the original theme, a sinuously chromatic melody with a somewhat clumsy closing cadence.
I had already learned the six-voice fugue when Doug’s 1977 GEB class began. Over the preceding winter break I learned the three-voice fugue as well. Not content with that accomplishment, I told Doug I would surprise him with a new canon on the Musical Offering theme. It was with great pleasure that I performed an original canon by augmentation (the faster voice plays the score twice in the same time as the slower voice plays it once) for the class. Here is the first voice.
Voice 1 of Canon by Augmentation on the Theme of the Musical Offering
Note that the first half of the composition is an embellished version of the original Musical Offering theme. On the next two pages are the full canon written out for two voices. Both voices start at the same time. The second voice is an octave lower and twice as slow, which means by the time the first voice ends, the second voice is only half done. To fill in the remaining time, the first voice then repeats, until the two voices catch up with each other at the very end (actually they end a bar or so shy of completion). Note that every note in the canon appears three times, and must play three different roles — a bit of musical algebra that I found quite absorbing to solve.
Some of Doug’s friends took my canon further, playing it in four voices with two people seated at one piano. The additional two parts play the theme even slower at quarter speed and eighth speed in lower octaves. The canon was not intended to work this way, so there are a few clashing dissonances, but the overall effect is quite entertaining.
Musical Offering Canon 2: by 50% delay at the octave
Years later I presented Doug with a second Musical Offering canon. In this brief eight-bar composition one voice starts four bars after the other, so the two voices are 50% out of phase with each other. I gave it to Doug originally as a “puzzle” canon, meaning I gave him one voice, and asked him how to transform the theme to produce a second accompanying voice. Here is the first voice of the canon.
And here is the full two-voice canon.
To compose this canon, I first played the theme against itself at the desired 4-bar delay. Here is the combined effect using the original theme without modification.
Rough first approximation of the 180° Canon on the Theme of the Musical Offering
The light gray areas of the canon work decently without modification; I added passing notes to give them more movement. The first two dark gray areas fall flat because both voices sit on the same note. To solve this problem I shifted some notes in time — in the first region I shifted the E flat in the bass voice earlier by a quarter beat, and introduced a scale-wise descent, so that by the time E flat sounds in the treble voice, the bass has moved down to a harmonious C. To enliven the chromatic descent in the third dark gray region, I delayed the treble voice by a quarter beat, creating a pleasing staggered pattern.
Because of the way this canon is structured, the last four bars of the piece are the same as the first four bars but with the voices reversed, so modifications to the first four bars work almost automatically in the last four bars. As I do when creating an ambigram, I created half the composition, then copied it to make the other half. In fact, the whole process of creating a canon is quite similar to creating an ambigrams: you start with a theme (word), choose a transformation (musical or geometric), modify the theme (word) so it both obeys the strict transformation and sounds (looks) harmonious (legible).
The finishing touch in composing both canons and ambigrams is to rationalize the quirks of the composition by making them part of a consistent style. In this canon, for instance, the alternating pattern of intervals in the first bar creates an angular marching rhythm that I continued throughout the piece. The goal is to compose a piece that works both mathemtatically and musically. I enjoy composing canons because the music seems to write itself, shaped by the pressures of the formal constraints.
Finally, here is a comparison of the original Musical Offering theme and the variations that appear in each of the canons, so you can hear how the theme has been altered.