To find how many strands are needed for a size 10 cable, I initially worked out the number needed for a size n cable. To do this, I considered the number needed for a size 1 cable, a size 2 cable, a size 3 cable, etc. This gave me the sequence (from a size 1 to a size 5 cable):
1, 7, 19, 37, 61
I then worked out the nth term, using the formula an^2 + bn + c. The difference between each value is 6, 12, 18 and then 24, and the difference between each of those differences is 6. "a" is therefore equal to 3 (6 / 2). I then worked out "c" by finding out the 0th term. The difference between the 0th and the 1st term is 0, as the difference between the differences is 6, so "c" is equal to 1. Finally, I worked out "b" by subtracting "a" and "c from the fist term. 1 - 3 - 1 = -3, so "b" is equal to -3.
The value for a size n cable is therefore 3n^2 - 3n + 1, or 3n (n – 1) + 1. I then tested that with a size 5 cable, which gave 75 - 15 + 1, equalling 61. The number of stands for a size 10 cable is therefore 300 - 30 + 1, which is 271.
Group 1’s approach makes the most sense to me. They split up the hexagon into three quadrilaterals, and then used the number of strands in each quadrilateral to work out the total number of strands. The first quadrilateral contains n^2 strands, the second contains n^2 – n strands and the third contains n^2 – 2n + 1 strands. They then added these toils together, to give the formula 3n^2 – 3n + 1, or 3n (n – 1) + 1.
I like the way the method breaks down the hexagon into three simple shapes, before combining the totals to give the overall number of strands. By breaking it down, it simplifies the method, but the hexagon is not broken down to such an extent that it overcomplicates matters, as in other methods.