April 2025
Welcome to EFM's April Newsletter!
It is essential that every caregiver in the world reads books and does math with their young children!
EFM believes in every child’s mathematical right to equity, opportunity, and personal fulfillment.
Celebrating EFM Volunteers
Our volunteers are essential for helping us reach people around the world. For the last six months I have been recognizing some of these wonderful people, and I would like to continue that tradition.
Russian – Danel, Violeta, Margarita, Viktoriia
A theme I hear a lot from these translators is that they want to give back to their communities, that they want to help improve math education for people of their native language. They know the advantages they had growing up in their households, and they want those experiences for all children.
Problem Solving Strategies
As I’ve written many times in these newsletters, problem solving is at the heart of doing math. It is not enough to expose your children to problem solving, it is important to give them tools that will help them flourish with it. Similar to any other aspect of learning mathematics, children benefit greatly from having guidance learning these strategies. If this training is started from a young age, by the time a child reaches middle school they will have a powerful set of tools they can wield to solve problems.
Singapore Math recognizes the importance of these strategies for young children. They have 12 strategies that they put into four groups. I like their emphasis on these strategies; however, I will present something simpler and I will focus on a few strategies that are particularly useful for younger children.
Puzzle – Bridges of Königsberg: I remember being given this puzzle in school. The puzzle asks whether it is possible to have a parade that travels over each of the bridges in the town of Königsberg exactly once. See the simplified map below.
It quickly became clear to me that it was impossible, but I had no idea how to prove it or otherwise explain the impossibility! I had no tools for looking into this and I was stuck. Let’s make sure the next generation does not repeat my experience.
Strategy 1: Learn from doing lots of examples, or from doing simpler versions of a problem, or from doing parts of the problem
There are many minor variations on this particular strategy, but this statement captures the gist of it. It is one of the most powerful and commonly useful strategies I know. My graduate studies advisor was a huge proponent of doing a mountain of examples and learning as much as you could from them. It is also a strategy that is easily practiced with children from a very young age.
For this puzzle, there are many simpler diagrams that one can play with, some of which are illustrated above. After playing with those, and perhaps others, a pattern emerges. Any node that has an odd number of sides attached to it must be the beginning or ending of a path. That is the key insight which leads to a complete understanding of these parade puzzles.
One example is often not enough to see the mechanisms at play in a puzzle. It’s too easy to get distracted by unimportant details in that particular situation. By looking at many examples, patterns emerge and common characteristics that are crucial start to stand out.
Another fun aspect of playing with examples is that they may lead you to ask new questions. After playing with these examples you might ask: For which diagrams can a path begin and end at the same point? For which ones is it impossible to have any path?
Puzzle – How to be a master at the game of Nim 1-2: I keep returnning to this game, because it is one of my favorite examples. Suppose you have ten things and you and your opponent are allowed to remove 1 or 2 things from the pile on each turn. The player who removes the last piece wins. Is it better to go first or second, and what is your strategy?
Ten is too big a number for seeing the strategy, so let’s reduce it to create simpler versions of the problem. When I asked my high school seniors to do this, they would choose numbers like 3 or 5. If it provides information, always start with the simplest version (1 in this case) and work your way up. Write down what happens if it’s your turn and there are 1, 2, 3, 4, or 5 things in a pile. In a few seconds the pattern emerges and you have conquered this game.
Puzzle – Are Their Hands Tied?: I have seen adults at parties struggle with this one for a long time. The puzzle says that the string is loosely tied around the wrists, so make a simpler version of this puzzle by making the space around the wrists quite large – almost immediately the solution is revealed!
Puzzle: Pirates with Gold: The usual version of this puzzle has five pirates and 100 pieces of gold, but I wanted a version more approachable to a 5-year old, and it is still challenging with just three pirates. Make this much easier by having just one pirate. Once you’ve solved it for one pirate, solve it for two, and then solve it for three. Notice that if you skip doing the one pirate solution, you still have some thinking to do for the two-pirate case – you should almost always do the simplest possible case first, especially if it only takes a few seconds to do.
Strategy 2: Keep good records – make organized tables, lists, and graphs, and keep intermediate calculations
I have had students do the simpler cases of the Nim game and still not see the pattern. What held them back was poor record keeping. They had their examples written all over their paper (or papers!), and so it was very hard for them to spot the pattern. Making organized tables, lists, and graphs doesn’t need to take long, and it can make all the difference.
Puzzle – Hallway: This puzzle is too sophisticated for the age group of this month’s newsletter, but I couldn’t resist it because it helps me make a point. Suppose you have a hallway with 1000 rooms all of whose lights are off. You have an automatic light switching system that runs amok – first it flips the switches in all the rooms, then it flips the switch in every second room, then in every third room, and so on until the last thing it does is flip the switch in every 1000th room. Which rooms have their lights on?
1000 is too big, so surely we want to learn from simpler versions. If you do just 1 or 2, that is too small to get anything useful, so this is a case where the very simplest versions are not a good idea. Try using 30 and seeing what happens. If you write the numbers from 1 to 30 in a line and keep track of the switchings, you’ll find that rooms 1, 4, 9, 16, and 25 are lit. You may recognize that these are all the squares and you will feel you’ve solved the problem.
But wait! Why are those the ones that are lit? If all you did was keep track of which were lit, you may have no idea. However, if under each number you wrote down which steps caused that flip to switch, you will see that those are the divisors of the number and that the numbers with an odd number of divisors are the ones that ended up being lit. It can be very useful to keep preliminary calculations and data – being tidy doesn’t mean throwing everything away!
In the September 2024 newsletter I talked about Data Analysis and the ways that tables, lists, and graphs can improve the understanding of a mathematical situation, so I will refer you to that article if you want more information on that topic.
Strategy 3: Look for extreme or isolated parts of the problem
Puzzle – Consecutive Numbers: Even very early puzzles benefit from these strategies. If you are placing the numbers from 1 to 8 in this grid, which of those numbers stand out? The beginning and ending numbers, 1 and 8, have only one neighbor, so they are unlike the other numbers (which have two neighbors). Should they be put in special locations? A bit of thought shows they must go in the center squares, and after you’ve done that the rest is pretty easy.
Puzzle – Sum Groups: Here is another early puzzle that benefits from these strategies. What are the extreme points here? The smallest and largest numbers are very restricted in how they can match, so those are good places to start. Also, the corners (and to a lesser extent the edges) are restricted. Once you’ve determined what happens with the 1s, 2s, 8s, and 9s, the rest of this puzzle goes very quickly.
Puzzle – Equal Products: The primes are often the distinctive numbers when it comes to multiplication. What happens to 2, 3, 5, and 7 here? The primes 2 and 3 can show up in other numbers from 1 to 9, but 5 and 7 only show up for themselves. So, 5 and 7 must be eliminated. After doing that, this puzzle becomes much easier.
Puzzle – Equal Sums: The extremes for this puzzle are the possible sums common to the three circles. There are three numbers in the middle circle, so the smallest the common sum can be is 1 + 2 + 3 = 6. Each of the outside circles has two numbers that they do not share, so the largest the common sum can be is 2 + 5 = 3 + 4 = 7. Now that the common sum is narrowed down to being 6 or 7, the puzzle is much easier.
Strategy 4: Use intelligent guess and check
There is a tendency for people to consider using Guess and Check as something an unintelligent person does when they have no ideas. Those people are needlessly losing the use of a very powerful strategy.
Puzzle – Keeping track of legs: A farmer has a total of 200 chickens and cows. Weirdly, the farmer counts the legs and notices there are 680 legs. How many of each animal are there, and what is wrong with this farmer?
If there were 200 cows, there would be 800 legs, and if there were 200 chickens, there would be 400 legs. Let’s guess 100 of each animal. That would give 600 legs. To get more legs, we need more cows. Let’s guess 150 cows – this gives 700 legs. Reduce the cows to 140 and you get exactly 680 legs.
The thoughtful person using Guess and Check here would notice that the number of legs increases by twice as much as the increase in the number of cows, and would then use that information to improve future trial numbers. If you are watching and thinking as you do Guess and Check, it has many of the strengths of Strategy 1.
Strategy 5: Identify a related problem you already know how to solve
There are many versions of Nim. Suppose you want to understand the version where you start at 0, add one or two things each time, and the person who brings the total number of things to ten wins.
This is just like the version of Nim described earlier, which we’ve already learned how to solve. If you think of the ten things sitting off to the side ready to be added to the pile, each time you add one or two to the adding pile you are removing one or two from the pile of ten – the adding pile will reach ten exactly when the subtracting pile reaches zero!
Strategy 6: Work backwards and also from both ends
This works very well for many geometry proofs, but of course those puzzles are more advanced than the focus of this newsletter.
Puzzle – Treasure Hunt: Even for a puzzle as small as this, there can be many choices to consider if we only start at the beginning and go forward. However, we can combine going forward with going backward from the goal.
Going forward from the Start we can go to either the upper right corner or the lower left corner. From those corners there are four places to consider, which begins to be a lot.
Which squares can land at the $$ goal square? There is only one square that can land there, which is the “1” in the square next to the $$. Continuing, there is only one square that can reach that “1,” and that is the “2” in the second row of the 1’s column. It will not be long before this backtracking reaches one of the four going-forward possibilities found earlier, and then you are done.
Wrapping Up
I hope you have enjoyed this discussion of problem solving strategies and exploring all those puzzles. For those looking for more, I recommend Polya’s groundbreaking (and old-fashioned) book “How to Solve It” and J. Mason’s excellent book “Thinking Mathematically.” If you are still wanting more, I have a rough draft of my own paper, “Problem Solving,” that you may find throught provoking. Find my paper at https://kitchentablemath.com/thoughts/problem-solving and download it by hitting the “Continue reading” button on that page.
If you have any questions or comments, please send them our way! We would enjoy the opportunity to chat with you. Also, if you are interested in collaborating with us or supporting us in any fashion, we would love to talk with you about ways we can work together!
April 18, 2025
Chris Wright
Chris@EarlyFamilyMath.org
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