\frac1{\sqrt2}({\ensuremath{|0\rangle}} - {\ensuremath{|1\rangle}})$. Initially, the two particles are in state $$\label{1}
 {\ensuremath{|\psi\rangle}} = \frac1{\sqrt3}\Big({\ensuremath{|01\rangle}}+{\ensuremath{|10\rangle}}-{\ensuremath{|00\rangle}}\Big).$$ That completes the description of the experiment. $\triangleleft$

In the rest of the section, we analyse the experiment. The initial state is given to us in the basis [$|00\rangle$]{}, [$|01\rangle$]{}, [$|10\rangle$]{}, [$|11\rangle$]{}. It will be convenient to express it in three additional bases. In the basis [$|0+\rangle$]{}, [$|0-\rangle$]{}, [$|1+\rangle$]{},[$|1-\rangle$]{}, we have $$\label{2}
\sqrt3{\ensuremath{|\psi\rangle}} 
= -{\ensuremath{|0\rangle}}({\ensuremath{|0\rangle}}-{\ensuremath{|1\rangle}}) + {\ensuremath{|1\rangle}}({\ensuremath{|+\rangle}} + {\ensuremath{|-\rangle}})
= -\sqrt2{\ensuremath{|0-\rangle}} + \frac1{\sqrt2}{\ensuremath{|1+\rangle}} + \frac1{\sqrt2}{\ensuremath{|1-\rangle}}.$$ In the basis [$|+0\rangle$]{}, [$|+1\rangle$]{}, [$|-0\rangle$]{}, [$|-1\rangle$]{}, we have $$\label{3}
 \sqrt3{\ensuremath{|\psi\rangle}} 
= {\ensuremath{|01\rangle}} - ({\ensuremath{|0\rangle}}-{\ensuremath{|1\rangle}}){\ensuremath{|0\rangle}} 
= \frac{\left({\ensuremath{|+\rangle}} + {\ensuremath{|-\rangle}}\right){\ensuremath{|1\rangle}}}{\sqrt2} - \sqrt2{\ensuremath{|-0\rangle}}
= \frac1{\sqrt2}{\ensuremath{|+1\rangle}} + \frac1{\sqrt2}{\ensuremath{|-1\rangle}} - \sqrt2{\ensuremath{|-0\rangle}}.$$ In the basis [$|++\rangle$]{}, [$|+-\rangle$]{}, [$|-+\rangle$]{}, [$|--\rangle$]{}, we have $$\label{4}
\sqrt3{\ensuremath{|\psi\rangle}} 
= \frac12{\ensuremath{|++\rangle}} - \frac12{\ensuremath{|+-\rangle}} - \frac12{\ensuremath{|-+\rangle}} - \frac32{\ensuremath{|--\rangle}}.$$

[Contextuality.]{} Again, suppose that we work with a noncontextual hidden-variable theory where the measurements are determined locally. Let $f_l(Z)$ and $f_l(X)$ be the sets of values that may occur as the result of the $Z$ measurement and $X$ measurement respectively on the left. Define $f_r(Z)$ and $f_r(X)$ similarly. By the noncontextuality assumption, these sets depend only on what happens on their side of the common source. But this leads to a contradiction.

Indeed, consider the 