= Solution
On these eight runs, $W=M$ and $SCT=-1$. The two independent defining words therefore give the full signed defining relation
$$
I=WM=-SCT=-SCWMT.
$$
Multiplying each desired <factorial contrast> by its four words gives all aliases. In the following equalities a minus sign means opposite contrast columns:
$$
\begin{aligned}
S&=SWM=-CT=-CWMT,\\
C&=CWM=-ST=-SWMT,\\
W&=M=-SCWT=-SCMT,\\
M&=W=-SCMT=-SCWT,\\
T&=WMT=-SC=-SCWM,\\
ST&=SWMT=-C=-CWM.
\end{aligned}
$$
Thus two <main effects>, $W$ and $M$, are inseparable even when every unwanted interaction vanishes. In addition, the important interaction $ST$ aliases with the <main effect> $C$. \b[Design (a) is unsuitable.] The seven-column <design matrix> for the intercept, the five main effects and $ST$ has rank five, confirming the two independent estimability failures.
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