Challenge Level

Imagine you have **two of each** of the 'weights' above.

Different combinations of the weights available allow you to make different totals.

For example:

$B + C = 6$

$B + 2C = 15$

$A + 2B + C = 4$

$2A + B + 2C + D = -10$

$B + 2C = 15$

$A + 2B + C = 4$

$2A + B + 2C + D = -10$

The largest total you can make is $20$ (check you agree).

The smallest total you can make is $-60$ (again, check you agree).

**Can you make all the numbers in between?**

Is there always a unique way of producing a total, or can different combinations produce the same total?

**Extension:**

If you are allowed just three different weights this time ($E$, $F$ and $G$), and at least one must be a negative weight, and you are now allowed to have up to **three of each,** you could choose:

$E = 1$

$F = -4$

$G = 5$

$F = -4$

$G = 5$

You could make $7$ and $-10$:

$E + F + 2G = 7$

$2E + 3F =$ $-10$

$2E + 3F =$ $-10$

Choose your three weights and test out which totals you can make.

**Which set of three weights ($E$, $F$ and $G$) allows you to make the largest range of totals with no gaps in between?**

*With thanks to Don Steward, whose ideas formed the basis of this problem.*