Understanding The Game of Odd Numbers by Paul Chika Emekwulu

112246/2 = 56123.

EXAMPLE 2

(a)  Start with any odd number say: 76125.

(b)  Add the next odd number to your number in (a): 

76125+76127=152252.

(c)  Add 2 to your sum in (b):

152252+2=152254.

(d)  Subtract 4 from your sum in (c):

152254-4 = 152250.

(e)  Divide your answer in (d) by 2: 

152250/2 = 76125.

Why the Trick Works

The above trick can be proved in two ways and verified using arithmetic.

We will therefore, explore the following:

(a)  Proof by Using Simple Algebraic Equations

(b)  Proof Using Expanded Notation

(c)  Using Arithmetic to Verify Our Results

Proof by Using Simple Algebraic Equations

Proof: (a)  Let us consider 2n+1 as an odd number.

The next odd number is (2n+1)+2 = 2n+3.

(b)  Adding the next odd number to our number in (a) we have:

2n+1+(2n+3)=(2n+2n)+(1+3)=4n+4.

(c)  Adding 2 to our sum in (b) we have:

4n+4+2=4n+6.

(d)}  Subtracting 4 from your result in (c) we have:

(4n+6)-4 = 4n+2.

(e)} Dividing your result in (d) by 2 we have:

(4n+2)/2=2n+1. 

 Mrs. Awosika: This is the original number we started with. How amazing!

If you took the general form of an odd number to be 2n-1.

Proof:  (a)  Let us consider 2n-1 as an odd number.

The next odd number is (2n-1)+2 = 2n+1.

(b)  Adding the next odd number to our number in (a) we have:

2n-1+(2n+1)=(2n+2n)+(1-1)=4n.

(c)  Adding 2 to our sum in (b) we have:   4n+2.

(d)  Subtracting 4 from your result in (c) we have:

(4n+2)-4 = 4n-2.

(e)  Dividing your result in (d) by 2 we have:

(4n-2)/2=2n-1. 

Mrs. Awosika:  This is the original number we started with.

Proof Using Expanded Notation

Proof  

Consider any three digit odd number abc.

Expressed in expanded form, abc = 100a + 10b + c. 

If abc is a three digit odd number, then the next three digit odd number is

(100a + 10b + c) + 2.

By addition, 100a + 10b + c + (100a + 10b + c) + 2 = 200a + 20b + 2c + 2

Adding 2 to the above sum:

(200a + 20b + 2c + 2) + 2  = 200a + 20b + 2c + 4

Subtracting 4 we have: (200a + 20b + 2c + 4) – 4

Dividing by 2 we have:

2(100a + 10b + c)/2 = 100a + 10b + c.

Mrs. Awosika:  Again, this is the original number you started with.

Using Arithmetic to Verify Our Results

EXAMPLE 1

(a)  Let us consider the number 343. 

343 = 300+40+3.

(b)  The next odd number is (300 + 40 + 3) + 2.

(c)  Adding  (a) and (b) we have: 

(300 + 40 + 3) + (300 + 40 + 3 + 2)

(d)  Adding 2 to our sum in (c) we have:

2(300) + 2(40) + 2(5)

(e)  Subtracting 4 from our sum in (d) we have:

=2(2000) + 2(300) + 2(40) + 2(3)

(f)  Dividing our result in (d) by 2 we have:

= 300 + 40 + 3  = 343.

This is the original number we started with. 

Even with arithmetic, the result is the same.

EXAMPLE 2

(a)  Let us consider the number 2421.  2421 = 2000 + 400 + 20 + 1.

(b)  The next odd number is (2000 + 400 + 20 + 1) + 2.

(c)  Adding (a) and (b) we have: 

(2000 + 400 + 20 + 1) + (2000 + 400 + 20 +1) + 2)

(d)  Adding 2 to our sum in (c) we have:

2(2000) + 2(400) + 2(20) + 2(3)

(e)  Subtracting 4 from our sum in (d) we have:

=2(2000) + 2(400) + 2(20) + 2(1)

(f)  Dividing our result in (d) by 2 we have:

=2000 + 400 + 20 + 1 = 2421.

This is the original number we started with. 

Even with arithmetic, the result is the same.

A Search for Counter Examples

Can you think of an odd number, call it x such that when you add the next odd number which is x+2, and add 2 to your previous result, subtract 4 and divide by 2, the result is not equal to your original number ?

This game works for even numbers just as it works for odd numbers.  This has been treated in a previous article. 

Mrs. Awosika: You will end up with the number you started with.

Oral Exercises

(a)  Can you think of a similar trick ?

Written Exercises

1.  How else can you prove that the trick discussed in this chapter works ?

2.  The sum of two consecutive odd numbers is 92.  Find the numbers.

3.  The sum of two consecutive numbers is 64.  If their difference is 1, find the numbers.

4.  Twice the sum of two consecutive odd numbers is equal to 32.  Find the numbers.

5.  The difference between two odd numbers is 18.  If their sum is equal to 48, find the numbers.

6.  The difference between two odd numbers is 14.  If the sum of two times the larger number and the smaller one is equal to 85, find the numbers.

7.  The sum of three consecutive odd numbers is 51.  If the sum of the first and the last is 34, find the numbers.

8.  Using a four digit number, verify algebraically that the trick discussed in this chapter works.

Mrs. Awosika:  “My name is Mrs. Awosika.  Good morning.”