4 x 4 x 4 Factorial Design or 4 x 4 Factorial with Blocking?

Problem here: https://imgur.com/Nl8AWie

I was thinking that this problem might just be a 3-factor factorial design, since obviously the ad campaign and packaging are factors (we're interested in their effects on sales), and the city where the investigation was carried out also seemed to be of interest since the cornflakes company is interested in testing the market; I was thinking that maybe if their product was a success in a certain city they'd want to know. Otherwise, this would just be a 2-Factor design with blocking.

The problem I have with this is in building the ANOVA table for the effects of the factors. The given data is just 16 responses, but for a 4 x 4 x 4 Factorial design the big N (or total sample size) should be 64. If I were to consider the 2-way and the 3-way interaction effects for this problem, I would have 63 degrees of freedom (df) for my Total Sum of Squares (SSTotal), and as it would turn out, I would have 0 df for my SSError, but I would have covered for the SS of all the effects. If I consider the sample size to be just 16, the df of my SSTotal would be 15, and it would be insufficient for the number of terms I have to include in my model. If I then consider a 4 x 4 Factorial design with Blocking, my SSTotal would have a large df = 63, assuming that my N should still be = 64, but again, the data is only 16 responses.

Am I thinking about this the wrong way, am I missing some vital lesson here? What do you guys think is the appropriate experimental design here? The designs that we were taught in class are completely randomized design (CRD), Random Complete Blocking Design (RCBD), Factorial Design (2-Factor, 3-Factor, 2^k Factor, 2^k with Blocking, Fractional Factorial).

I've been struggling with this particular problem for a couple of days now. This was part of our homework but I didn't manage to answer it, and my professor hasn't given us the answer yet (she probably really won't) and I need to know how to answer these kinds of problems by now since we have an exam tomorrow.

• While this is too late for your exam. This design appears to be a Latin square, randomized block designed experiment. – Dave2e May 6 at 15:18