Problem

The numbers listed below are to be packed into bins of size $n$, where $n$ is a positive integer.

$14 \quad 20 \quad 23 \quad 17 \quad 15 \quad 22 \quad 19 \quad 25 \quad 13 \quad 28 \quad 32$

A lower bound for the number of bins required is 4.

(a) Determine the range of possible values of $n$. You must make your method clear.

(b) Carry out a quick sort to produce a list of the numbers in descending order. You should show the result of each pass and identify your pivots clearly.

When the first-fit bin packing algorithm is applied to the original list of numbers, the following allocation is achieved.

Bin 1: $14 \quad 20 \quad 23 \quad 15$

Bin 2: $17 \quad 22 \quad 19 \quad 13$

Bin 3: $25 \quad 28$

Bin 4: $32$

When the first-fit decreasing bin packing algorithm is applied to the sorted list of numbers, the following allocation is achieved.

Bin 1: $32 \quad 28$

Bin 2: $25 \quad 23 \quad 22$

Bin 3: $20 \quad 19 \quad 17$

Bin 4: $14 \quad 13 \quad 15$

(c) Determine the value of $n$. You must explain your reasoning fully.