Help with Problem Investigating Structures Made of Unit Rods

[Carlito1]

Hey. I’m having trouble with this work I’ve been set about structures. I understand some of it, which I’ll explain later, but I really need your help with it!

My task is to investigate different sized cubes, made up of single unit rods and justify formulae for the number of rods and joints in the cubes. For example, 2 by 2 by 2 cube, 3 by 3 by 3 cube etc etc etc…

I desperately need help on getting a formula to work this out. I know already, as my teacher gave us these results, that a 1 by 1 by 1 cube has the following:

8 Three-Joints ( as do all cubes)
0 Four Joints
0 Five Joints
0 Six Joints

and a 2 by 2 by 2 has:

8 three joints
12 four joints
2 five joints
1 six joints

I really need a formula in order to calculate how many joints a 3 by 3 by 3 has, and with that formula to use it on let's say a 55 by 55 by 55 cube.

All help would be appreciated!

 

[Carlito1]

Sorry for being confusing, but here's the task:

Sorry, I was vague.
---
Rigid Structures are constructed using unit rods.

An example is shown below:

* Pic opf a 2 by 2 by 2 cube*

The individual rods in the structures are held together using different types of joints. (Just think of making cubes using matchsticks and playdough*.

http://img61.imageshack.us/img61/6166/joints3lz.jpg

A three joint would be on the vertices of each cube no matter what the size and so therefore there'll be 8 three-joints.

 

[Architeuthis Dux]

I would be happy to help you with your problem investigating structures made of unit rods. It seems like you are on the right track and have a good understanding of the basics. The key to solving this problem is to understand the pattern in the number of joints as the size of the cube increases.

Here's a possible approach that might help you come up with a formula:

1. Start by drawing out the 1 by 1 by 1 cube and labeling each joint with a number (1, 2, 3, etc.). This will help you visualize the pattern.

2. Then draw out the 2 by 2 by 2 cube and label each joint again. Notice how the joints from the 1 by 1 by 1 cube are still present, but there are also new joints.

3. Look at the new joints in the 2 by 2 by 2 cube and try to identify a pattern. For example, you can see that each edge of the cube has one new joint, and each face has four new joints.

4. Use this pattern to come up with a formula for the number of joints in a 2 by 2 by 2 cube. For example, you could say that the number of joints is equal to 8 (from the 1 by 1 by 1 cube) plus 12 (4 edges times 3 new joints per edge) plus 6 (6 faces times 4 new joints per face). This gives you a total of 26 joints.

5. Now, try to apply this same approach to a 3 by 3 by 3 cube. Draw it out and label each joint. Then look for the pattern in the new joints and use it to come up with a formula for the total number of joints.

6. Once you have a formula for the number of joints in a 3 by 3 by 3 cube, you can use it to calculate the number of joints in a 55 by 55 by 55 cube.

I hope this helps guide you in the right direction. Remember to always think critically and look for patterns when solving scientific problems. Good luck!