Math Mystery #1 The Checkerboard

While I gave you some pictures to ponder on the last post about what makes a set closed, I think it is time that I present you with a problem, I need to solve: The Checkerboard Problem.

The 8 by 8 Checkerboard

This is my travel checkerboard. It has A LOT of squares on it!

The Problem:  

How Many Squares Are On The Checkerboard?

 
 This morning, while I was at home with my dogs, I started thinking, "I wonder how many squares are on this checkerboard..." I decided to count them all and saw that there were 64 squares all together. I was happy with this until I noticed my dogs also became interested in the checkerboard.

**I should warn you now: After all the years of hanging out with me, my calculators and my math books, my dogs have become pretty good at math. They don't really like letting other people know about this, but when we are home alone, sometimes they help me out with some math problems!**

That's my ShihTzu, Champ. He was right... that big square should count.

 So now we are up to 65 squares - the 64 little square I counted and the one big square Champ found.

That's Buffy, she's a beagle. I thought about what she was thinking "ANY size square".

Problem Solving
 
I decided to stop for a moment and think about the question again and what I know about it so far...

• The Question: How many SQUARES are on the checkerboard?
• What I know about it (so far...):
• squares are four sided figures
• all of the sides have to be the same length
• each "space" on a checkerboard is a square (there are 64 of those!)
• Champ found one really big square. It is a square because every side is eight boxes long.
• Buffy thinks there are other sizes...

Do YOU know what Buffy is talking about? I think  have an idea, but it is going to take me some time to figure it all out.

I think we should try to figure this out. If my dogs can do this, then so can we! If you come up with a solution or have any questions, either post them in the comments section of this post, or e-mail me at blogwithnv@gmail.com. (ALL Math Daze Detectives that solve this Math Mystery will be spotlighted on our new MATH DAZE DETECTIVES PAGE.)

Good luck, have fun and happy counting!

TIP: I have an idea that might help: GRAPH PAPER. Have you ever seen this stuff?? It is paper covered in squares! We could use graph paper to make up as many checkerboards we need so we can count up all the different size squares that Buffy is thinking about without getting all confused. (If you don't have graph paper, you can print it out for free from one of my favorite websites MathBits.com. Just remember you need 8 by 8 squares to represent your checkerboards!).