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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.

  • 0
    @JasperLoy: I consider the first two equations equivalent because they will always return the same result, no matter what numbers I choose since `(((a−b)×c)/d)+e=(((a−b)/d)×c)+e`. While the fact that `(6−3)×2=(3−2)×6` relies on the particular choice of numbers. So at the end even if the same set of numbers is involved I consider the two solutions different. I'm not very good at explaining myself, but I hope to have been more clear, what do you think?2012-02-13
  • 0
    @RikPoggi, it seems that we just need to solve the equation(s) f(a,b,c,d,e)=g(a,b,c,d,e). Maybe there are some interesting things in your question, but that may be beyond me.2012-02-14

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