<s>[INST] $x$ as the shortest alternating sum of $1 \ldots n$ If I have an positive integer $x \in \mathbb{N}$ and I have $Z = \sum_{i = 0}^{n}{i}$ such that $Z \geq x$ and $Z - x \equiv 0 \bmod 2$ and $n$ is the smallest such integer it is possible to create and alternating sum from $1 \ldots n$ such that it equals $x$.
For example:
Say $x = 11$, then $Z = 15$ and $n = 5$ and the sum is $1 - 2 + 3 + 4 + 5$.
I see why the condition is necessary I don't see why it is sufficient.
 [/INST] Let $S(n)=\{x:x=\pm1 \pm2 \cdots \pm n\}$.  Let $T(n)=\{x:|x|\le \frac{n(n+1)}{2}\}\cap\{x:x\equiv \frac{n(n+1)}{2}\pmod{2}\}$.
Claim: $S(n)=T(n)$ for all $n\in \mathbb{N}$.
Proof: Induction on $n$.  $n=1,2$: clear
$S(n+1)\subseteq T(n+1)$: consider all $+$ or all $-$
$T(n+1)\subseteq S(n+1)$: $$[0,\frac{(n+1)(n+2)}{2}]\subseteq[-\frac{n(n+1)}{2}+n,\frac{n(n+1)}{2}+n]$$ and $$[-\frac{(n+1)(n+2)}{2},0]\subseteq[-\frac{n(n+1)}{2}-n,\frac{n(n+1)}{2}-n]$$
Now, let $ x\in T(n+1)$ and $x\equiv \frac{(n+1)(n+2)}{2}\pmod{2}$.  By the above pair of inclusions, either $x+n$ or $x-n$ is in $T(n-1)$.  But also $\frac{(n+1)(n+2)}{2}- n =\frac{n^2+3n+2- 2n}{2}\equiv \frac{n^2+n+2}{2}\equiv \frac{n(n+1)}{2} \pmod{2}$. [and $n\equiv -n\pmod{2}$].  Hence we're done by the inductive hypothesis.
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