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What is the minimum number of hyperplanes needed to divide the space of $\mathbf{R}^n$ into $m$ regions? I know that for $\mathbf{R}^2$, if we denote $f(m)$ to be the minimum number of lines needed to divide the regions, then we get $f(m)=f(m-1)+1+m-1$ by assuming that the new lines intersect with all previous lines. That line is cut into $m$ new segments and each segments generate a new region.

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