April 2023
Welcome to EFM's April Newsletter!
EFM: Supporting families to play, explore, and love math
More Math Puzzles to Enjoy!
This month I highlight the puzzles in James Tanton’s two books: Without Words and More Without Words. Each book contains 36 math puzzles that are described entirely without words, using only illustrations that the reader uses to figure out what the puzzle is. You can find some of these puzzles in various forms at his website, his project Global Math Project, and his Twitter @jamestanton.
One of the charming aspects of presenting puzzles without words or explicit instructions, beyond that they require no translation, is that this is the way mathematics is discovered. You discover a new pattern you think is promising by looking at lots of examples, deciding what you think should be true about them, and then setting about unraveling the puzzle or result.
I will refer to each puzzle by using a book reference, WW (Without Words) or MWW (More Without Words), followed by the puzzle number. Due to lack of space, I will be using words to describe these puzzles. You may recognize some of these as puzzles, or variations, we have added to the EFM Puzzle of the Week collection.
WW17 Suppose you have a rectangular grid of squares and a marked starting point. When is it possible to have one continuous path that starts at the starting point and then visits every square exactly once?
This first board is a 3 by 4 board with a starting position that works, and here is also a 3 by 5 board with a starting position that does not work. Which starting positions on these boards work? What happens for other sizes of boards? Can you describe some patterns in these results?
WW19,23,25 Last month I briefly described some puzzles that are associated with Fibonacci Numbers. Here are three of them. Spread out over time, each of these can provide a fun chance to do some organized counting and looking for patterns – all the better if your child already knows about the Fibonacci Numbers.
WW19 asks how many ways there are of going up 10 steps if you can do a mixture of climbing the steps one or two at a time. WW23 asks how many ways there are of filling a row of 10 squares if you can fill them one or two (neighboring) at a time. WW25 asks how many ways there are of filling 10 holes with stones if you aren’t allowed to put stones in neighboring holes (be sure to include the possibility of not using any stones).
WW24 If you have a collection of squares, when is it possible to exactly cover the squares with 1 by 2 dominoes?
The classic version of this puzzle involves removing one, two, or more squares from a chessboard. There is, of course, no solution if you have an odd number of squares. It gets trickier when you consider removing two squares from corners of the chessboard – sometime it is possible to cover it and sometimes it is not!
Of course, you can have more fun get creative making unusually-shaped boards of squares to cover.
MWW11 Suppose you start with 5 pennies that are heads up. If you choose any 3 coins to flip at each turn, you can flip all the pennies to tails in just 3 turns, as shown.
Suppose you start with 9 pennies that are heads up and you are allowed to flip any 5 pennies at each turn. What is the fewest number of turns it will take?
What happens for other combinations of pennies and flips? When is it possible to eventually flip them all and when is it impossible? When it is possible, can you predict the fewest number of turns it will take?
MWW20 Here are three examples of L-shaped pieces. The first one is a single piece. The second one is constructed from 4 smaller L-shaped pieces, and the third one from 9 smaller L-shaped pieces. Can you do this for 16 pieces? How about 25 pieces? How about other numbers that are squares? To be successful, must it be a square number?
This one is easiest to play with on a gridded piece of paper, such as graph paper. Alternatively, you can cut out L-shaped pieces and your child can move them around to see how they fit together.
This puzzle uses L-shaped pieces, but similar questions can be asked using other shapes of pieces.
If you have any questions or comments, please send them my way. I would enjoy the opportunity to chat with you. Also, if you are interested in collaborating with us or supporting us in any fashion, I would love to talk with you about ways we can work together!
Chris Wright
April 18, 2023
Chris@EarlyFamilyMath.org
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