Parity for Beginners
by rrusczyk, Jun 17, 2006, 3:21 PM
I'm finally back to writing about mathematical subjects. I'm putting together a few classes for a conference this summer for highly talented math students. Part of this project is writing a little material for the students (and their parents) to play with before the classes. This is the first in a series of four such articles. Most of these articles will be written primarily for the students to use on their own, but this will be written assuming parents (or older siblings) will work with the students. The material here is intended for students ages 5 through 8, though I think some of it will be of interest to older students, as well.
Parity: How Odd? Or Even?
One of the simplest concepts in math is odd or even. A number is even if it is evenly divisible by 2. For example, 14 is even because 14 divided by 2 equals 7, so 14 is divisible by 2. 3 is odd since we don't get a whole number when we divide 3 by 2.
Here's another way to look at it: if you have some number of objects that you can split into equal piles, then you have an even number of objects. For example, if you have 16 pieces of candy, you can split them into two piles of 8 pieces each. However, if you have 9 pieces of candy, there's no way to divide the 9 into two piles equally. Therefore, 16 is even and 9 is odd.
Here's a quick game. Identify whether each of the following is odd or even:
5
8
12
7
20
37
After you have a hang of odds and evens, try higher numbers, like:
46
77
80
94
57
At this point you might see the quick way to determine oddness and evenness:
Once you know how to say if a number is odd or even, try saying whether each of the following is odd or even:
4 + 7
8 + 10
12 + 7
11 + 13
6 + 5
13 + 10
Maybe you just added them and checked if the number was odd or even. If so, try this experiment - go back through the list above and say whether each number you are adding is odd or even, then look at the sum and see if it is odd or even. For example, for 4 + 7, you'll have
$\begin{eqnarray*} 4 + 7 &=& 11\\ \text{even} + \text{odd} &=& \text{odd} \end{eqnarray*}$
Now try to come up with a general rule for when a sum will be odd or even. Then say whether each of the following is odd or even:
23 + 45
44 + 38
103 + 676
684 + 21
44030 + 66531
Did you get these answers: Did you find the general rule?
Now that you're a pro determining if a sum is even or odd, try coming up with rules for subtraction or multiplication. Try some simple cases to find a rule, like these:
9 - 4
7 - 6
16 - 14
3 x 2
2 x 8
5 x 3
8 x 3
At this point, some of you may be wondering, 'Well, what's parity?' Parity is simply a short way of saying 'oddness or evenness'. For example, two numbers have the same 'parity' if they are both odd or both even. They have different parity if one is odd and the other even. So, we can shorten some of our rules above. We can say, the sum of two numbers with the same parity is always even, and the sum of two numbers with different parity is always odd.
Parity isn't just for adding, subtracting, and multiplication. Parity can be used to figure out all sorts of problems. Here are a couple simple games you can play to see how you can use parity to solve problems.
For each of the following games, we start with pennies arranged in a circle. On any turn, you can flip two adjacent pennies over. You can flip any penny over on as many turns as you like.
Game 1: Place 6 pennies in a circle such that two adjacent pennies are heads-up and the rest are tails-up. Try to get all pennies heads-up using the moves described above.
Game 2: Again, place 6 pennies in a circle. This time, make two pennies heads up and 4 tails-up such that there is one tails between the heads (i.e. HTHTTT). Again, try to make all the pennies heads in a series of moves as described above.
Game 3: Place 5 pennies in a circle, all tails-up. See if you can make them all heads in a series of legal moves (flipping two adjacent coins). Important: You can't stop halfway through a turn - you must always flip two coins! If you think it's impossible to make them all heads up, explain why.
Games 4 - ?: Vary the number of coins in initial arrangements, as well as the number of coins which start off with heads. See if you can find a general rule for when it is possible to make all the coins heads through a series of moves in which you flip over two adjacent coins each turn.
Here are a couple more puzzles that involve the idea of parity.
Puzzle 1: (For chess players) A knight starts at a square of a chessboard and makes a series of moves before returning to its original position. Is it possible that the knight made an odd number of moves?
Puzzle 2: I put the students in my class in a circle. There are 7 girls in the class, and I am able to seat them so that each student is either between two girls or between two boys. How many boys are in the class? (If you have trouble with this one, try it with 3 girls or 3 boys.)
Puzzle 3: Can the 5 x 5 grid shown be covered by 1 x 2 pieces like the blue one shown, such that no two of the 1 x 2 pieces overlap or go off the 5 x 5 grid?
Puzzle 4: One of the more famous problems in mathematics is the Bridges of Konigsberg problem. A river passed through the center of Konigsberg, Germany. An island sat in the middle of the river, and shortly after the island, the river split into two parts. As the image shows below, there were bridges connecting the island to either side of the river, and bridges where the river split. In all, there were 7 bridges, situated as shown. The famous problem is this: is it possible to go for a walk and cross each bridge exactly once (without going for a swim in the river!) For those going to the conference, we'll have a series of games leading up to the solution of this famous puzzle, which was solved by one of the greatest mathematicians of all time, Leonhard Euler.
Parity: How Odd? Or Even?
One of the simplest concepts in math is odd or even. A number is even if it is evenly divisible by 2. For example, 14 is even because 14 divided by 2 equals 7, so 14 is divisible by 2. 3 is odd since we don't get a whole number when we divide 3 by 2.
Here's another way to look at it: if you have some number of objects that you can split into equal piles, then you have an even number of objects. For example, if you have 16 pieces of candy, you can split them into two piles of 8 pieces each. However, if you have 9 pieces of candy, there's no way to divide the 9 into two piles equally. Therefore, 16 is even and 9 is odd.
Here's a quick game. Identify whether each of the following is odd or even:
5
8
12
7
20
37
After you have a hang of odds and evens, try higher numbers, like:
46
77
80
94
57
At this point you might see the quick way to determine oddness and evenness:
Once you know how to say if a number is odd or even, try saying whether each of the following is odd or even:
4 + 7
8 + 10
12 + 7
11 + 13
6 + 5
13 + 10
Maybe you just added them and checked if the number was odd or even. If so, try this experiment - go back through the list above and say whether each number you are adding is odd or even, then look at the sum and see if it is odd or even. For example, for 4 + 7, you'll have
$\begin{eqnarray*} 4 + 7 &=& 11\\ \text{even} + \text{odd} &=& \text{odd} \end{eqnarray*}$
Now try to come up with a general rule for when a sum will be odd or even. Then say whether each of the following is odd or even:
23 + 45
44 + 38
103 + 676
684 + 21
44030 + 66531
Did you get these answers: Did you find the general rule?
Now that you're a pro determining if a sum is even or odd, try coming up with rules for subtraction or multiplication. Try some simple cases to find a rule, like these:
9 - 4
7 - 6
16 - 14
3 x 2
2 x 8
5 x 3
8 x 3
At this point, some of you may be wondering, 'Well, what's parity?' Parity is simply a short way of saying 'oddness or evenness'. For example, two numbers have the same 'parity' if they are both odd or both even. They have different parity if one is odd and the other even. So, we can shorten some of our rules above. We can say, the sum of two numbers with the same parity is always even, and the sum of two numbers with different parity is always odd.
Parity isn't just for adding, subtracting, and multiplication. Parity can be used to figure out all sorts of problems. Here are a couple simple games you can play to see how you can use parity to solve problems.
For each of the following games, we start with pennies arranged in a circle. On any turn, you can flip two adjacent pennies over. You can flip any penny over on as many turns as you like.
Game 1: Place 6 pennies in a circle such that two adjacent pennies are heads-up and the rest are tails-up. Try to get all pennies heads-up using the moves described above.
Game 2: Again, place 6 pennies in a circle. This time, make two pennies heads up and 4 tails-up such that there is one tails between the heads (i.e. HTHTTT). Again, try to make all the pennies heads in a series of moves as described above.
Game 3: Place 5 pennies in a circle, all tails-up. See if you can make them all heads in a series of legal moves (flipping two adjacent coins). Important: You can't stop halfway through a turn - you must always flip two coins! If you think it's impossible to make them all heads up, explain why.
Games 4 - ?: Vary the number of coins in initial arrangements, as well as the number of coins which start off with heads. See if you can find a general rule for when it is possible to make all the coins heads through a series of moves in which you flip over two adjacent coins each turn.
Here are a couple more puzzles that involve the idea of parity.
Puzzle 1: (For chess players) A knight starts at a square of a chessboard and makes a series of moves before returning to its original position. Is it possible that the knight made an odd number of moves?
Puzzle 2: I put the students in my class in a circle. There are 7 girls in the class, and I am able to seat them so that each student is either between two girls or between two boys. How many boys are in the class? (If you have trouble with this one, try it with 3 girls or 3 boys.)
Puzzle 3: Can the 5 x 5 grid shown be covered by 1 x 2 pieces like the blue one shown, such that no two of the 1 x 2 pieces overlap or go off the 5 x 5 grid?
Puzzle 4: One of the more famous problems in mathematics is the Bridges of Konigsberg problem. A river passed through the center of Konigsberg, Germany. An island sat in the middle of the river, and shortly after the island, the river split into two parts. As the image shows below, there were bridges connecting the island to either side of the river, and bridges where the river split. In all, there were 7 bridges, situated as shown. The famous problem is this: is it possible to go for a walk and cross each bridge exactly once (without going for a swim in the river!) For those going to the conference, we'll have a series of games leading up to the solution of this famous puzzle, which was solved by one of the greatest mathematicians of all time, Leonhard Euler.
February 2012
January 2012