## Problem of the Day #106: Find Some Sums
*July 3, 2011*

*Posted by Albert in : potd , trackback*

Let $S = \{1, 2, \cdots, 9\}$. The function $f(x)$ maps $S$ onto some permutation, $P$, of $S$ for $x \in S$. For $x \not\in S, 1 \le x \le 9$, the graph of $f(x)$ is a straight line connecting the two nearest integer points.

Suppose $g(x_1, x_2)$ is defined as the area under $f(x)$ between $x_1$ and $x_2$, for $x_1 \le x_2$. Find the lexicologically minimal concatenation* of $P$ such that:

- $g(1, 9 ) = 43$
- $g(2, 8 ) = 38$
- $g(3, 7 ) = 29$
- $g(4, 6 ) = 16$

*If $P = \{1, 3, 2\}$ and $P = \{3, 1, 2\}$ are answers, the lexicologically minimal concatenation of $P$ is $132$.

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