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Steel Cables

Stage: 4 Challenge Level: Challenge Level:1

If you are a teacher, click here for a version of the problem suitable for classroom use, together with supporting materials. Otherwise, read on...

 

Cables can be made stronger by compacting them together in a hexagonal formation.

Here is a 'size 5' cable made up of 61 strands:


cable cross section, one cable in the middle with four rings of cables around it

How many strands are needed for a size 10 cable?

How many for a size n cable?

Can you justify your answer?

 


Once you've had a go at the problem, click below to see the diagrams some 15 year old students produced when they worked on it.
Do these diagrams give you any ideas for how you could work out the number of strands needed?

Group 1

student's picture of hexagon split into three quadrilaterals, two 5*5 rhombuses and a 4*4 rhombus

Group 2

student's picture of cable with horizontal arrows showing row lengths n, n+1, n+2 up to 2n-1 in the middle and then decreasing back down to n

Group 3

student's picture of hexagon split into six triangles

Group 4

student's picture of hexagon showing four rings and one cable in the centre


The work that these students did using their diagrams is given on the Getting Started page, if you would like a bigger hint.
 

Which of the four approaches makes the most sense to you?
What do you like about your favourite approach?

Can you think of any other approaches?

 

 

 

Notes and Background

Hexagonal packings are often chosen for strength or efficiency. To read more about packings, take a look at the Plus articles Mathematical Mysteries: Kepler's Conjecture and Newton and the Kissing Problem