1. Pathways
  2. Number Patterns and Infinity
  3. Noticing and Explaining Patterns
  4. Odd Square

Odd Square

 

Think of a number.
Square it.
Subtract your starting number.

Is the number you're left with odd or even?

Try with other numbers.

What do you notice?

Can you use the diagrams to explain what you have noticed?
 

                                                              odd_squares_3.gif

 

 

Think of a number.
Cube it.
Subtract your starting number.

Try it a few times and record your results.

What do you notice this time?

Is there a diagram or model that can help you explain what you have noticed?

 

 

Comments

Submitted by Jan S. Dybicz on Mon, 04/11/2016 - 08:44

Take any integer x, and raise it to some positive integral power n. Thus, we get x^n. If we subtract the original integer from the equation, we get x^n - x. An odd number raised to any positive integral power is going to be odd, and an even number raised to any positive integral power is going to be even. An odd number subtracted from another odd number is going to be even, and an even number subtracted from another odd number is also going to be even. This means that no matter the number, we will get an even number.

This only works for integers, though. If we start using non-integers like the golden ratio or 1/2, we could get values that are odd, fractional, irrational, or complex.

ajk44's picture

Submitted by ajk44 on Thu, 04/14/2016 - 15:11

That's a nice use of algebra and number properties to explain why the answer is even. I wonder if anyone can provide an argument that uses the image in the problem?