The riddle
What Side of the Turkey Has the Most Feathers?
The word side sets the trap
Asked "what side", a listener thinks left or right, front or back. Those are the sides of an object. This question wants the other meaning of side, the surface facing out as opposed to the inside. Feathers grow from the skin and cover the outside of a bird. Nothing grows on the inside, so the outside has them all. The riddle works by letting the listener pick the wrong meaning of one small word.
Left against right
A turkey, like most animals, is bilaterally symmetrical, meaning its left half is a mirror image of its right half (Britannica, "symmetry"). The feathers follow the same plan, so neither side could win the contest if it were a fair one. Anyone who starts counting on the left and then on the right will be puzzled to find a tie, which is the cue that the question means something else.
What a feather is
Britannica describes a feather as a light, flat growth from the skin of a bird, built around a central shaft with branches called barbs that lock together into a smooth surface. Among living animals, birds are the only ones that have feathers. Flight feathers, body feathers and soft down beneath all share that basic design.
Giving the group a turn
Ask the question and let each listener name a side. Left, right, top and back will all be offered. Hold the answer until someone says "inside", then ask which side that would favor. The reasoning lands more clearly when the group says it aloud.