= Solution
An \b[<orthogonal block design> has treatment contrasts <orthogonal> to block contrasts after removing the grand mean]. This is a statement about vectors on the <experimental units>, with the usual <inner product>; the two full spaces are not orthogonal because both contain the constant vector.
For a counting criterion, let $n_{ij}$ count <experimental units> receiving <treatment> $i$ in block $j$, let $r_i=\sum_jn_{ij}$, let $k_j=\sum_in_{ij}$, and let $n=\sum_ir_i$. The inner product of the centered indicators for treatment $i$ and block $j$ is $n_{ij}-r_ik_j/n$. Consequently
$$
\boxed{n_{ij}=\frac{r_i k_j}{n}\quad\text{for every }i,j.}
$$
In particular, equal-sized blocks must contain each <treatment> in the same proportion. Under an additive block-and-treatment model, adjustment for blocks then does not change the fitted <treatment contrasts>.
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