 Solution

On these eight runs, $W=M$ and $SCT=-1$. The two independent defining words therefore give the full signed defining relationMultiplying each desired factorial contrast by its four words gives all aliases. In the following equalities a minus sign means opposite contrast columns: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 $C$. 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.