This puzzle consists of a series of 24 arrowhead shaped pieces that are used to build a pentagonal icositetrahedron.
Each edge-edge connection is arranged so that each face's word abides by the other piece's edge rule.
Once the shape is built, it is time to look at the arrows located on some pieces. Each arrow or set of arrows points directly at another piece. There are eight faces pointed at, and the words they contain have something in common. They are the "fakes" from the flavortext and need to be "corrected". A "corrected" face will form a word when HEAD is appended, a condition the other sixteen "genuine" faces already satisfy.
The eight replacement words should be uniquely determined, as they also should match the five constraints from the five neighboring pieces.
Once these eight new words have been determined, the next step is to index into the new words by the number of the small arrows pointing at them. This gives one letter from each. By starting at the three red arrows, the result is the answer DOLLDOMS.