= Solution
Let $m=\overline Y_{+++}$, $r_i=\overline Y_{i++}$ and $c_j=\overline Y_{+j+}$. The proposed <least-squares estimators> satisfy the sum-to-zero constraints because $\sum_i r_i=Im$ and $\sum_j c_j=Jm$. Define $e_{ijk}=Y_{ijk}-r_i-c_j+m$. Direct summation gives
$$
\sum_{j,k}e_{ijk}=0\quad\hbox{for every }i,\qquad \sum_{i,k}e_{ijk}=0\quad\hbox{for every }j.
$$
Now write any other admissible parameters as $\mu=m+u$, $\alpha_i=r_i-m+a_i$, $\beta_j=c_j-m+b_j$, where $\sum_i a_i=\sum_j b_j=0$. Expanding the <ordinary least squares> criterion, the cross-products with $e$ vanish by the preceding identities. The cross-products between $u$, $a_i$ and $b_j$ vanish by their zero sums. Consequently
$$
S=\sum_{i,j,k}e_{ijk}^2+IJK u^2+JK\sum_i a_i^2+IK\sum_j b_j^2.
$$
Every added term is nonnegative and all vanish only at $u=0$, $a_i=b_j=0$. This proves the unique constrained minimum, with
$$
\boxed{\widehat\mu=m,\qquad\widehat\alpha_i=r_i-m,\qquad\widehat\beta_j=c_j-m.}
$$
The additive-model <residual sum of squares> and intercept-only <residual sum of squares> are therefore
$$
\operatorname{RSS}_1=\sum_{i,j,k}(Y_{ijk}-r_i-c_j+m)^2,\qquad \operatorname{RSS}_0=\sum_{i,j,k}(Y_{ijk}-m)^2.
$$
The same <orthogonality> gives the <analysis of variance> identity
$$
\operatorname{RSS}_0=\operatorname{RSS}_1+SS_A+SS_B,\qquad SS_A=JK\sum_i(r_i-m)^2,\quad SS_B=IK\sum_j(c_j-m)^2.
$$
Fitting factor B alone yields fitted values $c_j$ and residual sum $\operatorname{RSS}_0-SS_B$. Fitting A alone gives residual sum $\operatorname{RSS}_0-SS_A$. Adding B after A then reduces it to $\operatorname{RSS}_1$, a reduction of $SS_B$ again. \b[The reduction due to B is $SS_B$ in either order.] This is <balanced factorial orthogonality>; it generally fails for unequal cell replication.
There are $12$ observations. The additive model estimates $1+(3-1)+(2-1)=4$ independent parameters, so the missing <degrees of freedom> are \b[A: 2, B: 1, residuals: 8]. The individual requested residual sums are below.
To check for an <interaction>, extend the <normal linear model> to $\mu+\alpha_i+\beta_j+\gamma_{ij}$, with each row and column sum of $\gamma$ zero. Equivalently, fit the six unrestricted cell means. Its <residual sum of squares> is the within-cell sum
$$
\operatorname{RSS}_{\mathrm{cell}}=\sum_{i,j,k}(Y_{ijk}-\overline Y_{ij+})^2.
$$
The interaction adds $(I-1)(J-1)=2$ parameters; the cell-means residual has $IJ(K-1)=6$ <degrees of freedom>. Under the no-interaction <null hypothesis> and the independent equal-variance <normal> error model,
$$
\boxed{F=\frac{(2.6867-\operatorname{RSS}_{\mathrm{cell}})/2}{\operatorname{RSS}_{\mathrm{cell}}/6}\sim F_{2,6}.}
$$
The numerator and denominator arise from orthogonal <Gaussian> projections, proving the exact <F-test>. Reject for a sufficiently large value. The printed additive <analysis of variance> table does not supply $\operatorname{RSS}_{\mathrm{cell}}$, so it cannot determine this test statistic numerically; the replicated cell data would do so.
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