Given 5 different numbers ($\in \mathbb N$) in a specific brackets pattern like:
$\left(\left(\left(x_1 + x_2 \right) - x_3\right) \times x_4 \right) / x_5 = \text{result}$
Only the brackets are fixed, the numbers and the operators can be permutated at will.
And there must be all the four operators.
Does someone knows how could I find out if a permutation of the numbers and of the operators that gives the same result
- is an equivalent solution under the distributive, associative and/or commutative law
- or is a different solution that "just happens" to give the same result
For example:
$\left(\left(\left(6 - 3 \right) \times 2\right) / 1 \right) + 5 = 11 $
is equivalent to:
$\left(\left(\left(6 - 3 \right) / 1\right) \times 2\right) + 5 = 11 $
but it's different from:
$\left(\left(\left(3 - 2 \right) \times 6\right) / 1 \right) + 5 = 11 $
that gives the same result only with this particular choice of numbers: 2,3 and 6.