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Suppose there are $n$ data values $x_1<x_2<\ldots<x_{n-1}<x_n$,and I've found a partition number $k$, such that $$ \left|\frac{1}{k}\sum_{i=1}^k(x_i-\hat{\mu_k})^2-\frac{1}{n-k}\sum_{j=k+1}^n(x_j-\hat{\mu}_{n-k})^2\right| $$ is minimal. Here $\hat{\mu}_k$ is the mean of the first $k$ values, and $\hat{\mu}_{n-k}$ is the mean of the last $n-k$ values.
But is this optimal partition unique for any set of mutually different data values? How could I prove the uniqueness (or the opposite)? Furthermore, under what conditions would the solution be unique?

-- EDIT 1 --

@Glen_b has given a nice answer which let me notice that the original problem description is incomplete in some sense. The method to partition two datasets by minimizing their variance difference is a heuristic way to do binary classification, and it works in practice. So I'm thinking of the underlying theoretical aspects of the particular problem. In practice the data are affected by noise and will never distributed in some regular symmetric style. Now if I assume the data are generated by a mixture of two Gaussian distribution with equal variance, is it possible to prove that the method mentioned above generates a meaningful result?

Suppose there are $n$ data values $x_1<x_2<\ldots<x_{n-1}<x_n$,and I've found a partition number $k$, such that $$ \left|\frac{1}{k}\sum_{i=1}^k(x_i-\hat{\mu_k})^2-\frac{1}{n-k}\sum_{j=k+1}^n(x_j-\hat{\mu}_{n-k})^2\right| $$ is minimal. Here $\hat{\mu}_k$ is the mean of the first $k$ values, and $\hat{\mu}_{n-k}$ is the mean of the last $n-k$ values.
But is this optimal partition unique for any set of mutually different data values? How could I prove the uniqueness (or the opposite)? Furthermore, under what conditions would the solution be unique?

Suppose there are $n$ data values $x_1<x_2<\ldots<x_{n-1}<x_n$,and I've found a partition number $k$, such that $$ \left|\frac{1}{k}\sum_{i=1}^k(x_i-\hat{\mu_k})^2-\frac{1}{n-k}\sum_{j=k+1}^n(x_j-\hat{\mu}_{n-k})^2\right| $$ is minimal. Here $\hat{\mu}_k$ is the mean of the first $k$ values, and $\hat{\mu}_{n-k}$ is the mean of the last $n-k$ values.
But is this optimal partition unique for any set of mutually different data values? How could I prove the uniqueness (or the opposite)? Furthermore, under what conditions would the solution be unique?

-- EDIT 1 --

@Glen_b has given a nice answer which let me notice that the original problem description is incomplete in some sense. The method to partition two datasets by minimizing their variance difference is a heuristic way to do binary classification, and it works in practice. So I'm thinking of the underlying theoretical aspects of the particular problem. In practice the data are affected by noise and will never distributed in some regular symmetric style. Now if I assume the data are generated by a mixture of two Gaussian distribution with equal variance, is it possible to prove that the method mentioned above generates a meaningful result?

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Suppose there are $n$ data values $x_1<x_2<\ldots<x_{n-1}<x_n$,and I've found a partition number $k$, such that $$ \left|\frac{1}{k}\sum_{i=1}^k(x_i-\hat{\mu_k})^2-\frac{1}{n-k}\sum_{j=k+1}^n(x_j-\hat{\mu}_{n-k})^2\right| $$ is minimal. Here $\hat{\mu}_k$ is the mean of the first $k$ values, and $\hat{\mu_{n-k}}$$\hat{\mu}_{n-k}$ is the mean of the last $n-k$ values.   
But is this optimal partition unique for any set of mutually different data values? How could I prove the uniqueness (or the opposite)? Furthermore, under what conditions would the solution be unique?

Suppose there are $n$ data values $x_1<x_2<\ldots<x_{n-1}<x_n$,and I've found a partition number $k$, such that $$ \left|\frac{1}{k}\sum_{i=1}^k(x_i-\hat{\mu_k})^2-\frac{1}{n-k}\sum_{j=k+1}^n(x_j-\hat{\mu}_{n-k})^2\right| $$ is minimal. Here $\hat{\mu}_k$ is the mean of the first $k$ values, and $\hat{\mu_{n-k}}$ is the mean of the last $n-k$ values.  But is this optimal partition unique for any set of mutually different data values? How could I prove the uniqueness (or the opposite)? Furthermore, under what conditions would the solution be unique?

Suppose there are $n$ data values $x_1<x_2<\ldots<x_{n-1}<x_n$,and I've found a partition number $k$, such that $$ \left|\frac{1}{k}\sum_{i=1}^k(x_i-\hat{\mu_k})^2-\frac{1}{n-k}\sum_{j=k+1}^n(x_j-\hat{\mu}_{n-k})^2\right| $$ is minimal. Here $\hat{\mu}_k$ is the mean of the first $k$ values, and $\hat{\mu}_{n-k}$ is the mean of the last $n-k$ values. 
But is this optimal partition unique for any set of mutually different data values? How could I prove the uniqueness (or the opposite)? Furthermore, under what conditions would the solution be unique?

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