Wire cages

We can make 'wire cages' by using rods as the edges of cuboids. Can you work out what the volume of each wire cage will be? How many rods will you use?
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Problem

 
You may wish to look here at a 2D task which could be explored before this 3D task. 
 



Wire cages are used to contain large rocks in some building projects.



Here is a small version of a 2 by 3 by 4 wire cage cuboid, made with plastic straws.


 
Image
Wire cages
 
 

We can see 14 red rods of length 2 units, 12 blue rods of length 3 units and 10 green rods of length 4 units.

For all the challenges below, no rod is allowed to be longer than 6 units and each cage is always a cuboid. 



Challenge 1



The example above of dimensions 2 by 3 by 4 uses exactly 36 rods and has a volume of 24 cube units.

Find all the cages that have a volume of 20 to 25 cube units inclusive and use 40 to 50 rods inclusive.

Challenge 2



The example above of dimensions 2 by 3 by 4 uses exactly 36 rods.

Can you find a cuboid that uses exactly 28 rods?

Can you find all the cuboids that use exactly 28 rods?

 

Challenge 3



Find at least one pair of cuboids of different dimensions, that when constructed: 

- Use exactly 100 rods 

AND

- Have each dimension greater than 2 rods.


This sheet contains a copy of the task.