December 2021
Welcome to EFM's December Newsletter!
News
Update: Chapters 1 and 2 have more illustrations and two new Activities - enjoy our new look!
We hope you have a 2022 filled with many happy math times with those you care about!
New Activities to Enjoy!
This month we focus on Math Festivals. These are math events put on for children to see that math is a beautiful world to explore and have fun in. One of the major organizations providing math festivals is Julia Robinson Math Festival. JRMF supports organizations around the world who want to put on events for children to see how wonderful math is. During the last year, JRMF has worked hard to create turnkey programs so that schools, neighborhoods, and organizations can put on physical or virtual Math Festivals. Be sure to contact them if you are interested in putting on a math festival in your community.
Several Early Family Math Activities have come from JRMF in the past, and this month we introduce you to a few more. These activities provide a mathematical playground for children to play in and have adventures in.
Chapters 2 — Frogs and Toads
Sitting on seven rocks in a row are 3 frogs, and 3 toads. The 3 frogs start on the left-most rocks, and the toads start on the right-most rocks. A frog can only jump to the right and a toad can only jump to the left. A jump can only be to a neighboring empty rock or over a single animal to an empty rock. The goal is for all the frogs and toads to completely exchange places. If you don't have any frogs and toads handy, use coins that are Heads up for frogs and Tails up for toads..
In this illustration, the first row is the starting position, and the next two rows show possible first two moves..
If this proves too challenging, remind your child of the problem solving strategy of learning from simpler versions of the problem - simplify this challenge by having 5 rocks with 2 frogs and 2 toads, or even start with 3 rocks with 1 frog and 1 toad.
There are lots of natural variations to look at if you are enjoying this puzzle. What happens for n frogs and n toads on 2n + 1 rocks? What about n frogs and m toads on n + m + 1 rocks? What changes, if anything, if there is more than one empty space in the middle? Can you come up with some strategies for the minimum number of moves possible? How about investigating what happens on a two-dimensional board where the animals can jump in two directions.
Chapter 2 — Blue Dot Solitaire
Make a line of any number of dots. Some of the dots will be blue and the rest will be yellow. The goal for a given pattern of dots is to remove all the dots. The only way to remove a dot is to remove a single blue dot. When you remove a blue dot, any dots immediately to the right or left of it will change color (blue to yellow, or yellow to blue).
For the starting pattern shown, there is a way to remove all the dots (hint: not the way shown). Is there more than one way to do it?
As always, starting with six dots may be too many for your child to consider. Start with smaller puzzles and learn from them! Discover good strategies when there are three, four, and five dots. Is there a pattern to which ones can be solved? Given any row of six dots, is there a simple rule for deciding whether it is solvable? What happens for larger numbers of dots?
Chapter 3 — Gerrymandering
This is a timely puzzle in this time of battles over creating redistricting maps. Take a 5 by 5 square with 10 red squares and 15 blue squares. For a given coloring of the squares, the goal is to split the large square into five separate districts of five squares each so that three of those districts have at least three red squares - this will give red majority control in three districts even though red is far from being the overall majority. When creating districts, each square in a district must share at least one side with another square in the district, and the five squares of a district must be connected together. The first two of these puzzles are fairly easy to solve, whereas the third one looks impossible (is it?).
Play with different colorings and also with having 8, 9, 10, 11, or 12 red squares and see what happens.. For which colorings is it possible for red to win and when is it impossible? What is the smallest number of red squares so that red has a chance to win? As before, if you are having fun with this, try it with larger maps – say 5 by 7 with 7 districts and see what can happen.
Chapters 4 — Ladybugs
You have a collection of ladybugs flying in the air, each of which has a unique number of black dots on its red back. One possibility is to have ladybugs that are numbered consecutively starting at 1. Suppose there are two leaves for the ladybugs to land on. They start landing starting with number 1. If at any time there are three ladybugs on one leaf where the numbers of two of the ladybugs adds up to the third, then they all fly away and you must start over.
What are the most consecutive ladybugs starting at 1 that you can have land on two leaves? How does this change if you start with the number 2? How about larger starting numbers? How does this change if you have three leaves, four leaves, or more leaves? What happens if you only have odd-numbered ladybugs? What happens if you only have even-numbered ladybugs? What happens if you use only multiples of some other number (e.g. 3, 6, 9, 12, ...)?
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
December 18, 2021
chris@kitchentablemath.com