This site will be hosting the 25th edition of Math and Multimedia Carnival which will go live at the end of this month, October 31.
A math blog carnival is a collection of articles from various math blogs and sites. So if you are a blogger, this is an opportunity for you to promote your favourite or latest posts and yes, your blogs for free. Below is a collection of blog carnivals I previously hosted.
If you have articles about math problems, puzzles and games, tips for teaching math and specific topics in math, videos, tutorials, lessons, curriculum materials and book reviews, math trivia especially about the the number 25, etc, you may email the permalinks to me or use the math and multimedia blog carnival submission form.
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This is the second in my series of posts in combinatorics. The first post links the Fundamental Counting Principle, Powers of 2, and the Pascal Triangle. This second post connects the Pascal’s Triangle and the formula for counting the number of permutations with identical objects. The context for connections is a puzzle about counting the total routes of a rook to squares in a chessboard.
The Puzzle
Trace the shortest route the rook can traverse from its corner position to the opposite corner. How many such routes are there?
Solution 1
A useful problem solving strategy here is to simplify the problem: Count the number of distinct paths the rook will land to the nearest square.
To determine the shortest path, the rook should either go north or east or left or right only. That is, the rook has only two choices each time it moves through a square as shown in the figure below. You will not reach the third or fourth line before you figure out that the pattern generates the Pascal’s Triangle.
Thus there are 3432 distinct shortest paths the rook can traverse from one corner to the opposite corner.
Solution 2
Another way to solve puzzle 2 is shown below. The arrows trace some possible shortest paths for the rook.
The paths of the arrows in the figure can be represented this way: (N = North, E = East)
N N N E N E E N E N E N E E
E E E E N N N E E N N N N E
E E E E E E E N N N N N N N
Each shortest path from one corner square to the opposite corner square is made up of 7 N’s and 7 E’s. Thus, counting the total number of shortest paths is the same as counting the total number of distinct arrangements of 7 N’s and 7 E’s in a row. The formula for counting the permutation of n objects a, b, c, … of which are identical will be useful here. This is given by the formula: . So,
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Click here to see solutions to this problem using the multiplication principle and combination formula.
If you want to practice your skill in solving permutation and combination problems, Examrace Permutation & Combination Made Easy (Examrace Aptitude Series) will be useful.
This post shows how we can help students make connections among counting principle, the Pascal’s triangle, and powers of 2. I have tried this lesson in an in-service training program but I’ve yet to test it with students in high school. The lesson uses the strategy Teaching thru Problem Solving.
A piece of knowledge is powerful to the extent to which it is connected to other piece of knowledge. The more connections there are, the more powerful it becomes. Mathematics teaching therefore should always aim to help students make connections among the different concepts of mathematics. You may want to read my article about understanding as making connections.
The Problem: Trace the paths that will spell “MATHEMATICS” starting from the letter M on top moving only downwards, either to the immediate letter to its right or to the immediate letter to its left. How many different paths are there in all?
After a few minutes and the class is seem getting nowhere you may suggest to students to try simpler case first like trying the word MATH. Trying simpler case is a good problem solving strategy and habit students need to learn.
Solution 1
Suppose we spell the word “MATH” only. From M we can move downwards and may either choose the A at the left or the A at the right. Having chosen an A we can either choose the T down left or the T down right. And having chosen one we can either choose the H down right or the H down left. Each time we only have two choices. Thus, the number of ways of tracing the word “MATH” in the above figure is
2·2·2 =23=8
Using the same line of thinking, the total number of paths which spells “MATHEMATICS” is
2·2·2·2·2·2·2·2·2·2=210=1024.
Solution 2
Notice the number of arrows that converges to a particular letter. It tells the number of paths that pass through it. Thus, to count the number of ways of tracing the word “MATH” we only have to add the total number of arrows that point to the H’s. There are
1 + 3 + 3 + 1 = 8.
Count the number of arrows converging to each letter in MATHEMATICS . You will generate the triangular array of numbers below.
The solutions showed two important principles of counting.
The Multiplication Principle. If one task can be done in m ways and then another task can be done in n ways, the pair of tasks, first one and then the other, can be performed in
m • n ways.
The Addition Principle. If one task can be done in m ways and another task in n ways, then one task or the other can be done in
m + n ways.
Anyone who wants to understand permutations, combinations and anything that involves counting should first understand these principles.
The triangular array of numbers generated above is one of the most influential number patterns in the history of mathematics. It is called Pascal’s triangle after the renowned French mathematician Blaise Pascal (1623-1662) who discovered it. The triangle is also called Yang Hui’s triangle in China as the Chinese mathematician Yang Hui discovered it much earlier in 1261. The same triangle was also in the book “Precious Mirror of the Four Elements” by another Chinese mathematician Chu-Shih-Chieh in 1303.
The Pascal triangle yields interesting patterns and relationships. Some of the obvious ones are:
To generate the next row, you will have to add the two numbers above it.
Another striking property of this array of numbers is its symmetry. Note the numbers on both sides of the middle number in each row.
The sum of the numbers in each row can be expressed in powers of two.
This site will be hosting the 25th edition of Math and Multimedia Carnival which will go live at the end of this month, October 31.
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will be useful.
