= Solution
Let $\mathcal U=(U_0,\ldots,U_m)$ be a finite <affine open cover> of the <projective scheme> $X$. For a <coherent sheaf> $\mathcal F$, form the <Čech cochain complex>
$$
\check C^p(\mathcal U,\mathcal F)=\prod_{i_0<\cdots<i_p}\Gamma(U_{i_0}\cap\cdots\cap U_{i_p},\mathcal F),\qquad
(\delta c)_{i_0\ldots i_{p+1}}=\sum_{j=0}^{p+1}(-1)^j c_{i_0\ldots\widehat{i_j}\ldots i_{p+1}}\big|_{U_{i_0}\cap\cdots\cap U_{i_{p+1}}}.
$$
The two ways to omit any pair of indices have opposite signs, giving $\delta^2=0$. Its <cohomology groups> are
$$
\check H^p(\mathcal U,\mathcal F)=\ker\delta^p/\operatorname{im}\delta^{p-1}.
$$
A <projective scheme> over a <field> is a <separated scheme>. Finite intersections of its affine opens are affine, and a <coherent sheaf> is <quasi-coherent>. Higher <sheaf cohomology> of a quasi-coherent sheaf on an <affine scheme> vanishes. Thus this is an acyclic cover and the <acyclic cover theorem> identifies the displayed <Čech cohomology> with $H^p(X,\mathcal F)$. In particular its degree-zero kernel glues compatible local sections to $\Gamma(X,\mathcal F)$. A cover by $m+1$ opens also gives vanishing for $p>m$.
For the <cohomology of twists on projective space>, take $S=k[X_0,\ldots,X_r]$ with the usual grading and $U_i=D_+(X_i)$. On an intersection indexed by the nonempty set $J$,
$$
\Gamma(U_J,\mathcal O(n))=\bigl(S[X_j^{-1}:j\in J]\bigr)_n.
$$
This follows directly from the construction of the <twisting sheaf on projective space>: sections of the twist are the degree-$n$ homogeneous fractions in the corresponding <localization>. The <Čech differential> preserves each <Laurent monomial> $X^a=X_0^{a_0}\cdots X_r^{a_r}$, where $a_i\in\mathbb Z$ and $\sum_i a_i=n$. Such a monomial appears precisely in summands with
$$
N(a):=\{i:a_i<0\}\subseteq J.
$$
The complex therefore decomposes as the direct sum of finite-dimensional <cochain complexes> indexed by these exponent vectors. Their differentials have only the alternating signs of the simplex incidence maps.
If $N(a)=\varnothing$, this is the ordinary unaugmented <simplex> cochain complex: its degree-zero kernel consists of the common value on every vertex and has dimension one, while all higher cohomology vanishes. If $\varnothing\ne N(a)\ne\{0,\ldots,r\}$, choose $v\notin N(a)$. A <cochain homotopy> inserting $v$ into the ordered index list, with the sign of that insertion, contracts this subcomplex. Insertion or deletion of $v$ preserves the condition $N(a)\subseteq J$; in $\delta h+h\delta$, terms inserting and deleting different vertices cancel in pairs, while the term inserting then deleting $v$ is the identity. Thus this entire monomial subcomplex is acyclic. Finally, if all $a_i<0$, the monomial occurs only in the full intersection, in degree $r$, and contributes one copy of $k$ there.
For $r\ge1$, these three cases give the complete answer:
$$
\boxed{H^i(\mathbf P_k^r,\mathcal O(n))\cong
\begin{cases}
S_n,&i=0,\ n\ge0,\\
\displaystyle\bigoplus_{\substack{a_0,\ldots,a_r<0\\a_0+\cdots+a_r=n}}k\,X_0^{a_0}\cdots X_r^{a_r},&i=r,\ n\le-r-1,\\
0,&\text{otherwise}.
\end{cases}}
$$
In the first case counting nonnegative exponent vectors gives $h^0=\binom{n+r}{r}$. In the top case write $a_i=-1-b_i$ with $b_i\ge0$; then $\sum b_i=-n-r-1$, giving
$$
\boxed{h^r(\mathbf P_k^r,\mathcal O(n))=\binom{-n-1}{r}\quad(n\le-r-1).}
$$
This is the <Laurent-monomial description of top cohomology on projective space>. All intermediate degrees vanish for every twist, and the top degree vanishes when $n\ge-r$. If $r=0$, then $\mathbf P_k^0=\operatorname{Spec}k$ and every twist is trivial, so \b[$H^0(\mathbf P_k^0,\mathcal O(n))=k$ for every integer $n$, with all higher groups zero].
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