Posts tagged ‘maze’
There is debate as to whether Carl Banks, creator of Scrooge McDuck, or Christopher Thomas originally coined the following phrase.
Work smarter, not harder.
Regardless of the author, it seems a good quote to keep in mind on Labor Day, and I was reminded of it when my sons were solving mazes yesterday. I thought they might enjoy learning a little theory that would make solving a maze easier, now that they’ve gotten quite good at completing mazes with brute force.
Before sharing with you what I shared with them, here are two maze-related jokes.
The first is a maze for liberal arts majors, who always seem to have trouble finding their way.
The second is a maze I call Good Frickin’ Luck.
With those out of the way, on to the theory of mazes.
For simply connected mazes — that is, those mazes for which every wall is connected to either another wall or to the boundary — solving the maze can be accomplished by coloring the walls with just two colors. That’s because such mazes consist of two disjoint parts, and the solution path lies between those two parts.
Consider the simple maze below, which, by the way, I created “on the spot” for my sons when they had completed all the mazes in their activity books.
You can then color all of the connected walls on the top half of the maze red, and you can color the other walls blue. This gives the following result, and the solution is the path that lies between the red and blue halves, as shown.
My kids thought that was pretty cool. Hope you did, too!
The summer is a great time for kids to hike, bike, swim… and forget everything that they learned during the school year.
The son returned to school after summer break. At the end of the first day, his mother received a call from the teacher about his poor behavior. “Now, just one minute,” said the mother. “He had poor behavior all summer, yet I never called you once!”
In Outliers, Malcolm Gladwell purports that poor kids lose ground to affluent kids during summer break. Their experiences and academic progress during the school year are similar, he contends, but their out-of-school experiences during the summer are very different. Though minor at first, the cumulative effect of those summer losses becomes noticeable as children get older.
The following are five games/puzzles that can be used with young kids to prevent summer losses and, possibly, even elicit some summer gains. Each has the characteristics that I love about a good game for young kids: It requires students to use and practice basic skills, but there is a higher purpose for doing so.
This is a game that’s kind of like SuDoku, but a million times better. If you don’t know the game, check it out at www.kenken.com. My sons noticed me playing it one afternoon and asked what it was. I explained, and they asked if they could do it with me. We now solve three or four games every afternoon. I used to help them a lot, but now they pretty much know all of their math facts up through 7 × 7. How do you not love a game that helps four-years-olds learn the times table?
This is a puzzle, not a game, and you can learn all about it at Math Pickle. The general idea is that you start with a sequence of numbers in a flower-like pattern. You then multiply two adjacent numbers, subtract 1, and divide by the number below. The cool and surprising part is that every intermediate result is an integer, so there are no ugly decimals for kids to deal with. And by the twelfth ring of petals, every result is 0. Happens every time.
3. Squares of Differences
The good folks at Math For Love reminded me of this great problem, and Josh Zucker discussed it at length on the NYTimes Numberplay blog. Draw a square, and put a positive integer at each vertex. Then at the midpoint of each side, write the difference of the numbers at the two adjacent vertices. Now connect the midpoints to form a rotated square inside the original square, and repeat. It seems that if you continue this process long enough, you’ll eventually get all 0’s. But does that always happen?
By the time kids test this conjecture with three or four attempts, they’ve done a hundred subtraction problems without even realizing it.
4. Decimal Maze
The Decimal Maze (PDF) comes from the lesson Too Big or Too Small on Illuminations. Trying to obtain the maximum value while traversing a maze with decimal operations, students learn about the effects of multiplying and dividing by decimals that are greater or less than 1. The activity is good for upper elementary and middle school students, but I’ve used modified versions with very young kids. For instance, a modified maze for kids in first grade uses single-digit positive integers while limiting the operations to just addition and subtraction; for older kids, a maze could include fractions or powers instead of decimals.
5. Dollar Nim
As I mentioned in a previous post, my wife created a great game that I call Dollar Nim. The idea is simple. Imagine you have 100¢, and on your turn you can remove 1¢, 5¢, 10¢, or 25¢. Players alternate turns; the player to reduce the amount to 0¢ is the winner. The optimal strategy is not obvious, and kids practice a whole lot of subtraction, especially as it relates to making change.
More generally, any one-pile nim game is great for the purpose of having kids practice subtraction without realizing it.
I hope you find some free time this summer to enjoy these games. I’ll leave you with a joke/truth about summer school.
I never understood the concept of summer school. The teacher’s going to go up there and go, “OK, class. You know that subject you couldn’t grasp in nine months? Well, we’re going to whip it out in six weeks.” – Todd Barry