Blocks
The 2k different treatments in a complete 2k factorial experiment can be split into two blocks by treating the blocking variable as an extra factor in the design and confounding it with a high-order interaction between the factors of interest.
If it is assumed that no interactions exist between the blocks and any of the k factors in the experiment, then all main effects and other interactions between the factors can still be estimated — they are only aliased with block-factor interactions.
Factorial design for 3 factors in 2 blocks
Each row on the left of the diagram below describes one of the eight treatments in complete design for three 2-level factors. The 'factor' D defines how the treatments are split into two blocks.
Initially D is confounded with the main effect of A so all treatments with low A are in block 1 and those with high A are in block 2. (The treatments in each block are also listed on the right.) This is a poor design since the main effect of A cannot be estimated or tested.
Click immediately above the heading for the ABC interaction to confound D with ABC. The column of -1 and +1 for the ABC interaction then define which of the treatments are in the two blocks. Note that the main effects and 2-factor interactions between A, B and C are confounded with interactions involving D, but if we assume that the blocks do not interact with the three factors of interest,
All main effects and 2-factor interactions between A, B and C can be estimated and tested.