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The Development of Algebra - 2

This is the second article in a two part series on the history of Algebra from about 2000 BCE to about 1000 CE.

3D Stacks

Stage: 2 and 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

This activity has been particularly created for the most able. (The pupils that you come across in many classrooms just once every few years.)

This activity also follow on for those who have successfully worked at Doplication. 
 
Let me help you visualise this representation of a 3D situation.
It's kind of two set of three adjacent cubes with balls at each vertex.
Towards the bottom left of the picture are the eight large balls, then going backwards there are eight medium red balls and further back are eight small ones.
In the centre of the front three cubes are three orange balls and at the centre of the rear cubes are three brown ones.

 
 new6
In Doplication you would have had;
 
 axitate version
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.