In the current circumstances of staying home during the Covid-19 pandemic, I’ve spent a lot of time doing puzzles. Some are jigsaw, some are pencil & paper, and all of them have got me thinking about the math ideas that the puzzles generate.
We have a 1000-piece rectangular jigsaw puzzle going on our kitchen counter. We begin by searching for edge and corner pieces (a method recommended by Jennifer Fairbanks @JenFairbanks8 in her puzzle post here.)
I started to wonder how many pieces would be along the long and short edges of the rectangle, and then… how many different ways can 1000 pieces be arranged in a rectangle? It turns out my puzzle had 40 pieces along the long edge and 25 pieces along the short edge.
How many edge pieces are there (single-edge) and how many corner pieces? What math did you use to count them? I figured it this way:
- There are 4 corner pieces
- Long side of 40 pieces – 2 corner pieces = 38 edge pieces
- Short side of 25 pieces – 2 corner pieces = 23 edge pieces
- Edge pieces = 2*38 + 2*23 = 122
Then there must be 874 interior pieces: I can subtract corners and edges from 1000, or multiply 38 long rows by 23 short rows to figure this out.
Another math-y feature of the puzzle is that its dimensions are 29 inches by 20 inches. I wondered about the approximate area of each piece.
Along the long edge, 29 inches ÷ 40 pieces = 0.725 inches/piece; along the short edge, 20 inches ÷ 25 pieces = 0.8 inches/piece.
How can we calculate the approximate area of each piece in square inches? [I like to use inch2 as a more mathematically meaningful unit when working with students.]
Even though 0.725 * 0.8 is hard to compute in my head or by hand, I’ve learned how to expand my thinking with Pam Harris’s weekly #MathStratChat on Twitter (Pam tweets at @pwharris) .
Here are some ways I think about this calculation. First, I believe that “Fractions Are Your Friends” so I use fractions instead of decimals. This method uses the strategy of decomposing 8 into its factors 2*4 and then multiplying 725 by these friendly numbers.
Another method is to go back to the fractions that originally created the decimals. I divided by the common factor of 20 (I try to avoid the wording “cancel out”). Next, I multiply by 2/2 which is a fraction with value = 1, so it changes the form of my fraction without changing the value.
A third method is to use a ratio table to multiply 725 * 8. I find annotations helpful to understand the math thinking behind the table. I needed to subtract to finish my table, and so I had two alternative strategies for computing 72 – 14: first, I “removed friendly numbers” and second, I “made the first number nicer and then compensated with an inverse operation.”
A final method I considered was to try to multiply the decimals directly (yikes!) by using a “double-and-half” strategy. Since doubling (multiplying by 2) and halving (dividing by 2 or multiplying by ½ ) are inverse operations, they undo each other. I used an additional inverse operation in the last step to help me think about multiplying by 0.1; I find this to be a more mathematically meaningful approach than “move the decimal”.
And YES! If each puzzle piece has area = 0.58 inch2, then the 1000 piece puzzle has area of 580 inch2 which makes sense for dimensions of 20 inch x 29 inch.
For those teachers who ask students to convert between units, I use a “unit analysis” structure. If we know that there are 2.54 cm in 1 inch, do we multiply or divide? This technique uses the concept that a fraction whose numerator and denominator are equal has a value of 1. So both of these fractions have the same value, and we choose appropriately to create the units we need (and cancel away the units we don’t need).
This result is close to the area if we had multiplied the centimeter dimensions given, but not exact due to rounding off (another good discussion to have with students!)
Some of our finished puzzles and work-in-progress:
Jigsaw puzzles aren’t my only puzzle-y way to pass the time at home! There are many, many opportunities to engage with math-y puzzles online, and here are some I’ve found fun:
1. Matt Parker’s Math Puzzles: the “stand-up mathematician” (@standupmaths on Twitter) is posting video puzzles every week or two at http://think-maths.co.uk/maths-puzzles and then solution videos within a few days. The puzzles are fun and accessible, and each one has taken me some time to play around with it. I love that there are many ways to solve and there isn’t an immediately obvious solution path. The team at Think Maths is keeping score, with points for correct solutions, bonus speed points, and partial credit for wrong answers. Here are some of my solution props:
2. I do quick daily puzzles from The Times Mindgames and Sunday Times Brain Power and @YohakuPuzzle on Twitter. These don’t require much time commitment and are great fun! Catriona Shearer (@CShearer41) writes geometry problems on Twitter, and many of her puzzles rely on simple and common geometry principles. Here are two recent ones I’ve solved:
3. It is very satisfying to play around with physical objects, and I’ve been exploring puzzles with Jenga blocks and Panda Squares, both posed by David K. Butler (@DavidKButlerUoA). David’s full set of Jenga Views are available here.
4. And finally, there are plenty of people and websites that are posting daily or weekly puzzles, or have archives of puzzles available. Here are some that I have encountered:
- Alex Bellos sets a puzzle on alternate Mondays in The Guardian: Monday Puzzle.
- Chris Smith (@aap03102 on Twitter) is posing “Corona Conundrums” on YouTube.
- Dan Finkel (@MathForLove) has all his puzzles here.
- Art of Problem Solving has puzzles on their website.
- Project Euler has a huge problem archive available, registration is free.
- Zach Wissner-Gross (@xaqwg) posts “The Riddler” at FiveThirtyEight.com.
So if you are looking for something to pass the time, get puzzling!
The puzzles shown are Pomegranate ArtPiece Puzzles. They are high quality and we are enjoying the variety of artwork they depict. The puzzles pictured above are:
- Frank Lloyd Wright: Saguaro Forms and Cactus Flowers, 1928
- Diego Rivera: Detroit Industry, North Wall (detail), 1933
- Birds & Flowers: Japanese Hanging Scroll
More can be found at Pomegranate’s website and at many online retailers.
Jigsaw puzzles have become a popular pastime during the pandemic, and have been hard to find at stores and online merchants. The New York Times published an article April 8, 2020 about the buying boom and the manufacturing process. [Photo credit: Roderick Aichinger, NYTimes]
Have you ever opened a jigsaw puzzle and found two or more pieces still connected? What do you do? Dave Richeson (@divbyzero) found the definitive answer with this Twitter poll: