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!).

The Counting Numbers... Naturally

The first numbers you ever began to discover were those that came naturally to you, the ones that allowed to realize that there were different amounts, or quantities, of things in the world. As your mind started to see these patterns emerge, someone in your life, probably a parent started to teach you about counting. If you were lucky, you were even taught how to do this in different languages. As you put together the patterns in your head that came naturally to you, with the numbers your loved ones taught you were used for counting, you probably became a counting monster!

I always find it fun to be around a child that just learned to count!

During this time you can learn about a lot of numbers and one fun activity is to see how high you can go, but (here's the really cool part) there is no end! You can go higher forever!

These numbers that you first learn as a child and then continue to use for the rest of your life have a name. They are called (this is a really great name if you ask me) the COUNTING NUMBERS. They are also known as the NATURAL NUMBERS. Both of these names make perfect sense to me because they are the numbers that we kind of already know naturally, but they are also the numbers we use for counting. So what numbers are they, exactly?


A Daze-y Definition

The set of Counting Numbers, or Natural Numbers, includes all of the non-decimal numbers (known as whole numbers - we'll get to that in a later post) greater than zero.  See the symbol and set below:

= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11...}

 It is very important to note that ZERO is NOT A COUNTING NUMBER. Zero is a very complicated number (even though it is all about nothing!), so it is something we will address in a later post. Just remember that the counting numbers begin with one and go up from there.

Questions to ponder... and then COMMENT on:

• What's the highest counting number you know?
• How many languages do you know the counting numbers in?
• What things can we do in life with counting numbers?
• What things in life need MORE numbers than just the counting numbers? (That seems crazy... I'm pretty sure I said these numbers would go on forever!)
• Do you remember who first taught you your counting numbers? (Have you thanked them yet???)
• Are you currently being tortured by a counting monster (aka - a child who just learned to count and, therefore must count EVERYTHING in sight)? 

Art created using the Drawing Online website.