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'Steel Cables' printed from http://nrich.maths.org/

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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 how some 15 year old students worked on it.
Can you explain their reasoning?

Group 1

student's picture of hexagon split into three quadrilaterals, two 5*5 rhombuses and a 4*4 rhombus table showing dimensions of the three quadrilaterals and total T for a size 2 up to 10 cable and a size n cable T=n^2+n^2-n+n^2-2n+1 = 3n^2-3n+1 or 3n(n-1) +1

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 student's method is adding up pairs of rows to make n pairs of (3n-1), with the 2n-1 row counted twice, giving a total 3n(n-1) +1

Group 3

student's picture of hexagon split into six triangles student's method is to use the formula for the (n-1)th triangular number n(n-1)/2, multiply by six, and then add 1 for the cable in the centre

Group 4

student's picture of hexagon showing four rings and one cable in the centre 1, 6*1, 6*2, 6*3, 6*4 ... 6*(n-1). We noticed that the area of each ring followed this pattern. To find the total we needed to add the area of each ring. 1 + 6*1 + 6*2 + 6*3 ... 6(n-1)=1+6(1+2+3+...+(n-1) 1+6(n(n-1)/2) 1+3n(n-1)

 

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