# Number sequences - drawing a diagram

• October 23rd 2005, 10:38 AM
Euclid Alexandria
Number sequences - drawing a diagram
From the book: "Draw a diagram that shows how the sequence of odd numbers can be used to produce the sequence of squares. The diagram can contain numbers, addition symbols, equal signs, and arrows, but it should not contain any words. Your goal is to draw the diagram so that anyone who knows how to add whole numbers can look at it and see the relationship between the numbers in the two sequences."

1, 3, 5, 7,... The sequence of odd numbers
1, 4, 9, 16,... The sequence of squares

Here is my answer. What do you think? Is this a clear diagram? Would you have done it differently?
• November 15th 2005, 12:25 PM
The Pondermatic
Good job!
I think that the diagram is very clear. I looked at the diagram and understood it before I read the post.
• November 21st 2005, 11:49 PM
Euclid Alexandria
Thanks!
• November 22nd 2005, 12:30 AM
CaptainBlack
I think the question is looking for something like:
• November 22nd 2005, 07:58 PM
Euclid Alexandria
Bear with me because that diagram looks like it should make more sense to me than it does. If I'm interpreting it right...well, examine the flow of my arrows as they go from point A to the solution. In the first block, 1+3=4, makes sense. In the next block, the odd number we need is on the outside bottom left, but in the third block, it's in the inside bottom right.

Wait, I'm just now seeing how it's all lined up on the top row in the third block.

I'll attach a second image.

That second block looks a little confusing, but just follow the arrow from the 4 to the 5 to the answer (9), to the 7 to the answer (16, which appears in the expanded third block). I just don't think I'd be able to figure it out at a glance, especially without the key on the bottom. It seems to act like more of a math puzzle (a fun one, but still)?
• November 22nd 2005, 09:25 PM
CaptainBlack
Quote:

Originally Posted by Euclid Alexandria
Bear with me because that diagram looks like it should make more sense to me than it does. If I'm interpreting it right...well, examine the flow of my arrows as they go from point A to the solution. In the first block, 1+3=4, makes sense. In the next block, the odd number we need is on the outside bottom left, but in the third block, it's in the inside bottom right.

Wait, I'm just now seeing how it's all lined up on the top row in the third block.

I'll attach a second image.

That second block looks a little confusing, but just follow the arrow from the 4 to the 5 to the answer (9), to the 7 to the answer (16, which appears in the expanded third block). I just don't think I'd be able to figure it out at a glance, especially without the key on the bottom. It seems to act like more of a math puzzle (a fun one, but still)?

Think of the blocks of numbers in the diagram as having blobs
in place of the numbers, then the corresponding number in the
diagram is the count of blobs around the diagram to that point.

The blobs all form square arrays and so have a total number of
blobs which is a square. And we see that at each stage we are
adding the next odd number of blobs as a border to the previous
diagram.

The diagram might have worked better without the numbers? :(

RonL
• November 23rd 2005, 05:56 PM
Euclid Alexandria
From your explanation, I now understand your diagram as a process of steps toward each sum, rather than a series of calculations around the matrix, which I knew wasn't its intention beforehand (even at first glance, I could tell 1 + 2 + 3 + 4 was not the diagram's intended message). I couldn't see at first glance that a process of steps was involved.

Yes, this diagram would make more sense without the numbers -- which was tempting; I could make some cool blobs with Photoshop. Except the book only wanted the diagram to contain numbers, addition symbols, equal signs, and arrows. So we're stuck with the numbers. The diagram makes sense after your explanation, but since no explanations can be used, we want a diagram that bodes well for first glances...not just for myself, but for a very broad audience: "anyone who knows how to add whole numbers," who "can look at it and see the relationship between the numbers in the two sequences."
• January 11th 2006, 08:09 AM
earboth
Hello,
I'm not quite sure if you are looking for that kind of diagramm, which I've attached below. I didn't invent it - I just found the right book.
• January 17th 2006, 04:17 AM
Euclid Alexandria
Second part makes sense, but first part less clear
Those are interesting diagrams. I also think it is a good example of what CaptainBlack was trying to describe with his diagram. Is there a page of text accompanying your diagrams? I understand some parts of it, but some parts of it are difficult to understand -- particularly the equations surrounding the first diagram, such as 1/2n(n+1), 2n+1, etc. I understand that n=5, but for instance, if 2n+1=11, I don't understand how that answer relates to the rest of the diagram.

Unfortunately, I don't think I could have used symbols and squares such as these to illustrate my diagram. The book only wanted the diagram to contain numbers, addition symbols, equal signs, and arrows. Furthermore, the diagram needed to be understandable for a very general audience, "anyone who knows how to add whole numbers" according to the book, and "anyone who can look at it and see the relationship between the numbers in the two sequences." If more symbols were allowed, perhaps my attached diagram would be clearer?
• January 17th 2006, 11:01 AM
earboth
hello, here I am again:

The first diagram shows a rextangle. The area contains exactly three times the sum of squares. The area of a rectangle is calculated as

$A=length \times width$

The long side contains the sum of integer:

$1+2+3+...+n=\frac{1}{2}\cdot n\cdot(n-1)$

the width measures 2n+1

So you get:

$3\cdot (1^2+2^2+3^2+...+n^2)=\frac{1}{2}\cdot n\cdot(n-1)\ \cdot \ (2n+1)$

that means:

$(1^2+2^2+3^2+...+n^2)=\frac{1}{6}\cdot n\cdot(n-1)\ \cdot \ (2n+1)$

And to the text which explanes those diagrams: It exists, OK, but it's in German and I don't know if that will help you further on.

I hope I was a little bit of assistance.

Bye