January 2023
Welcome to EFM's January Newsletter!
News
Progress Please check out some of our new materials. Our Activities for Educators program now has 70 Puzzles of the Week and 35 Math Games for the Classroom. Our Korean translator has been busy and has translated the first three chapters of games, puzzles, activities, and investigations.
Donations EFM received two donations just before the end of the year. One for $1000 and another for $250. The second one will be matched by their employer in a couple weeks. We look forward to using those funds to produce playing cards, reading books, or illustration tools. Thank you all for your support!
The World is a Mathematical Playground!
Recently, I spent time with my two granddaughters, Zoe (1) and Claire (3 ½). I got to see the world through their eyes for a while. Claire had so many interesting questions, and each answer and observation led to more questions. She was discovering patterns and wanting to learn more. To a young child, the world is a mathematical playground.
We at EFM want the world to feel this way for all children of all ages.
Lately, I have been busy happily improving EFM’s collection of Puzzles of the Week, which is part of the Activities for Educators program we are developing. We have been adding a section to each puzzle called “Exploration,” and also a teacher’s section of “Notes” discussing the solution and associated ideas of each puzzle.
Some of these puzzles naturally lead to new questions and new puzzles to solve. When a math puzzle creates a mathematical playground like that, I like to call them Rich Math Environments. Dan Finkel calls them Rich Tasks, and has a section of the Math for Love website on them.
Encouraging a child to look beyond the immediate puzzle and consider associated questions of their own is a wonderful thing. It pulls them into interesting and deeper aspects of the material, it moves them away from being passive consumers of mathematics and gives them ownership over what is being thought about, and it moves math further away from right and wrong answers and into an exciting world of exploration and discovery.
There are several excellent books that look at this process. One I’m currently enjoying is The Art of Problem Posing by Brown and Walter. Their problems are for an older age group, but their strategies for eliciting problem posing are universal. Another excellent book is Thinking Mathematically by Mason et al., which explores a large collection of interesting problems. A third book is Measurement by Paul Lockhart, which is a lovely journey of discovery through geometry, often seeing what can be posed and discovered simply by presenting a simple diagram and nothing else. By the way, if you have not read his A Mathematician’s Lament, you really should look it up.
Let’s Play in this Playground!
Filling Squares with Squares
In how many squares can you cut a big square into? A simple question with lots of interesting results!
After some thought, the first answer is the square numbers: 1, 4, 9, 16, and so on. This is already a good realization for a child to have, and represents an insight into breaking a square into pieces of equal size.
How can we adjust this question or understand it more thoroughly? What if we allow the squares to be of different sizes? What happens if we replace all the squares in this question with some other shape? What happens if we change the dimension of the problem?
If we allow different sizes of squares, we now have a lot more options. When flooded with options, it is helpful to find ways to work systematically, which is a good habit to promote in children. While unstructured play with this problem is great stuff and should be encouraged, eventually, perhaps a few years from now, your child will need to look for structure in the answers.
One very helpful observation is that any solution can replace a single square to create a new solution. That is what was done in the third example – the bottom right square was replaced by four squares. The net effect of replacing one square by four is to increase the total number of squares by 3. By adding three to all our old answers, perhaps repeatedly, we get 1, 4, 7, 9, 10, 12, 13, 15, 16, 18, 19, and so on.
There is at least one other systematic construction to look at, but I won’t spoil all the fun.
What happens when squares are replaced with other shapes? For which other shapes would it even be possible to do that? Lots of good fun here! Certainly we can decompose regular triangles into smaller regular triangles. Surprisingly, the story for regular triangles is similar to that of squares. Do other shapes work and lead to interesting results? Parallelograms … Hexagons … ? Would it be interesting to have more than one kind of shape for the pieces?
What happens when you break a cube into smaller cubes? That sounds like fun, but I haven't tried it.
Ladybugs that don’t Add up or Multiply
Suppose we have numbered ladybugs landing on a leaf. The one requirement is that a leaf cannot have two ladybugs whose numbers add up to a third ladybug on the same leaf. The leaf on the left is okay, but the one on the right is not because 2 + 4 = 6. Using two leaves, how high can you get starting at 1?
Some fiddling around suggests that {1, 2, 4, 8} and {3, 5, 6, 7} are as good as it gets. That was fun and lots of good practice with addition and number bonds was had.
How can we play with this situation a bit further? One way is to change the kinds of numbers we work with – instead of using consecutive numbers starting with 1, we can use even numbers, odd numbers, prime numbers, or even Fibonacci numbers. Another direction we can change in is how many leaves there are. What happens if there are 3, 4, or more leaves? What if we change the combination rules? What happens if we require that no two numbers can multiply to give a third on a leaf? A simple problem has blossomed into so many possibilities!
Let’s look at some of these.
Using even numbers turns out to be not a big change. If we multiply all the numbers in our first answer by 2, we get the best answer for even numbers. Not every change will lead to an exciting result, though observing the usefulness of scaling by 2 is a good experience for a child. For odd numbers, something very surprising happens – the numbers can all be put on one leaf! I’ll leave primes and Fibonacci numbers for you to play with.
With more than two leaves, it gets hard in a hurry. It’s so hard that it is an unsolved problem what the best result is for six leaves.
Still, there are some things we can play with and get some interesting results. Notice that there are at least two kinds of groups that can always be put on a single leaf. One is the powers of 2, such as {1, 2, 4, 8, 16}. The other is that we can put all the consecutive numbers starting at a number and going up to its double on a single leaf, such as {3, 4, 5, 6}. Those two ideas turn out to be more useful than you might think.
To get a very good answer for three leaves, start with the answer for two leaves {1, 2, 4 8} and {3, 5, 6, 7}, and use {9, 10, 11, 12, 13, 14, 15, 16, 17, 18} on the third leaf. You can then start at 19 and sprinkle those numbers where you can. After some playing around, you arrive at the answer {1, 2, 4, 8, 22}, {3, 5, 6, 7, 19, 20, 21}, and {9, 10, 11, 12, 13, 14, 15, 16, 17, 18}. That is only one away from the best answer of 23, and you can arrive at that answer by making two small changes in this preliminary answer!
This general topic is called sum-free sets and sum-free partitions. If you look up those results, you’ll see that the answers in the literature when adding one more leaf are similar to what happened when going from two leaves to three. This leads to a conjecture: Does the best answer for one more leaf always include the previous answer inside it?
Suppose we change the problem from addition to multiplication? Now we are requiring that two numbers on a leaf cannot multiply together to give a third number on that leaf. Something very interesting with primes and prime factorizations happens. One leaf can have all numbers that are 1, a prime, or a prime squared. The other leaf can have the remaining numbers up to where it fails. That looks like {1, 2, 3, 4, 5, 7, 9, 11, 13, …, 47} and {6, 8, 10, 12, 14, 15, 16, … 46}. I suspect that’s the best answer for two leaves. What happens for three or more leaves? What other questions can you think of?
Pan Balances
You are challenged to weigh any possible weight from 1 ounce to 30 ounces on a pan balance. What is the smallest set of weights that will work?
For a child seeing this for the first time, the answer is not obvious. So, encourage your child to suggest a smaller problem to work with. Perhaps replace 30 with 5, then 6, 7, 8, 9, and 10. With a few examples, the answer will become clear.
To find more to investigate, examine the assumptions. Did your child assume that all of the weights would be on one side of the device? If so, there’s a completely new puzzle that opens up letting the weights be on both sides.
What happens if we start with some limited types of weights and ask which things can be weighed? Suppose we have an unlimited supply of 4 ounce and 7 ounce weights. Which things can be weighed and which cannot? What happens if we replace 4 and 7 by other pairs of numbers? For which pairs of numbers are there an unlimited collection of things that cannot be weighed? For pairs where that doesn’t happen, can you discover a formula for where the cutoff is? Can you discover other properties of the set of things that can’t be weighed?
Again, you get an entirely new problem if you allow your 4 ounce and 7 ounce weights to be on either side of the device. You can ask all those questions again, and they have interesting answers!
Light Switch Problem
You have a long corridor of 100 rooms. All the rooms are dark. You have 100 crazy robots. The first robot goes down the corridor and flips every light switch. The second robot flips every second light switch. The third robot flips every third switch, and so on. After all the robots are done, which rooms will be lit and which will be dark? There are many equivalent forms for this question, such as using locked lockers in a locker room.
This is one of my favorite rich math environments, because it can lead to so many interesting number theory questions and also it is a good training ground for problem solving techniques. It is a bit advanced for EFM, so I’ll only briefly touch on it.
The initial question can be settled fairly straightforwardly with a bit of patience. The lit rooms are the ones numbered 1, 4, 9, 16, …, 100, which are exactly the squares. If you stop there, you have answered the question but missed a lot of fun math.
Why is it that the squares are lit and what is really going on? It turns out, a room’s light switch gets flipped for each one of the divisors of the room number! So, a room will end up lit if it has an odd number of divisors. From our result, we conjecture that a number has an odd number of divisors exactly when it is a square. Wow! Why is that?
Now we are led to exploring how to count divisors. Which numbers have each of 1, 2, 3, 4, and 5 divisors, and are they interesting (they are)? What is interesting about each group? Is there a systematic way of counting the divisors for any number? Are there some counts of divisors that force a number to be of a certain form?
Counting divisors was fun. What happens if we add up a number’s divisors? What happens if we multiply together a number’s divisors? What other interesting questions can you come up with for this problem?
What if we do the original problem with crazy robots who only flip odd numbered switches? What if they are only even numbered switches? Multiples of 3 … primes … numbers from 1 to 10 … Not every variation will have an interesting answer, but you will have a fun time finding out!
What a Wonderful Mathemtical World!
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
January 18, 2023
Chris@EarlyFamilyMath.org
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