# Ordering four real numbers such that some conditions are satisfied

Suppose I have 4 real numbers in $$[0,1]$$, all different between each other.

Assume that there exists a way of ordering the four numbers such that: $$\begin{cases} w_1>w_2\\ w_3=1-w_1\\ w_4=1-w_2 \end{cases}$$ where $$w_1$$ is the first number in the ordered sequence, $$w_2$$ is the second number in the ordered sequence, $$w_3$$ is the third number in the ordered sequence, $$w_4$$ is the fourth number in the ordered sequence.

Is such a way of ordering unique? If yes, could you sketch the proof? In not, could you give a counterexample?

• What exactly do you mean when you say unique? – Ankit Kumar Dec 20 '18 at 13:31
• can you give such an assignment to the following four numbers? $\frac{1}{4},\frac{1}{8},\frac{1}{16},\frac{1}{32}$ – user408906 Dec 20 '18 at 13:38
• @Suraj: I'm assuming that such an assignment exists. My assignment requires that you can construct two pairs summing up to 1. In your example, this is not possible. – STF Dec 20 '18 at 13:41
• What about $0,\frac{1}{4},\frac{3}{4}, 1$ and $\frac{1}{4},1,0,\frac{3}{4}$? [I choose to do $w_1<w_2$ and swap $w_3$ and $w_4$ it's the same thing.] – ancientmathematician Dec 20 '18 at 13:52
• No. Think about it. – ancientmathematician Dec 20 '18 at 13:59