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Consider a $ 2^3 $ factorial design lay out in 2 blocks ,each of size 4, as follows

Block I: {1,a,b,c}

Block II: {ab,ac,bc,abc}

Here,the treatments combinations are written in Yate's notation. Then which of the following are always true?

a) Main effect A is confounded

b) Main effect B is unconfounded

c) Interreaction AB,BC,AC are all unconfounded

d) Interreaction ABC is confounded

below I show my attempt

considering the following expression while confounding ABC

(a-1)(b-1)(c-1)

the possible division is

Block 1: (1 ,AB ,BC ,CA)

Block 2: (A , B , C , ABC )

There must be something wrong with the question or I am missing something ? Please suggest if I am wrong .

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You are correct. I too feel that there should be some discrepancy with the problem. Normally we avoid confounding main effects, as we don't want to lose information on them. The effect which is confounded has an even number of letters in common with all the elements of the controlled block (Here the block containing treatment combination $(1)$). You have already shown a possibility where $\mbox{Int} ABC$ has been confounded. Similarly, if we would like to confound $\mbox{Int} AB$ then the treatment combinations are as follows:-

Block I: $(1),(ab), (c), (abc)$

Block II $(a),(b),(ac),(bc)$

Under the assignment given in the particular problem you could not confound any of the first and second order interactions.

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