3D Stacks

Can you find a way of representing these arrangements of balls?
Age
7 to 14
Challenge level
filled star filled star filled star
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem

This activity may be used as a follow-on for those who have successfully worked at Doplication.

 

Let me help you visualise this representation of a 3D situation.

In this example of a 2 by 3 by 4 we see in the red balls, as two layers of 3 by 4.

 
3D Stacks
These two layers together can also be viewed as representing 6 cubes.


3D Stacks
 

 
In the top picture the centre of each cube is shown with a light/dark brown ball.

 


In Doplication you would have had;
 
 
3D Stacks
which we called $4$ ? $3$.

 
In this activity it would be called something different involving $4, 3, 2$. Just for now I am going to use @, so the diagram above represents $4$@$3$@$2$.

 
I invite you to explore many such arrangements and find a way of recording your results.
You might like to just look at the arrangements that are like cubes - $3$@$3$@$3$ , $4$@$4$@$4$, $5$@$5$@$5$, etc.
 
As with most mathematics it's good to compare so maybe compare the results for squares in Doplication and cubes in 3D Stacks.