Two Groups Share an Odd Number
The number of Innocents in the overlap between two named groups is odd (1, 3, 5…). If the intersection is a single cell — as it always is when a row and a column cross — this reduces to: that cell is an Innocent (since 1 is odd and 0 is even). For larger overlaps, valid counts are any odd values from 1 up to the size of the intersection. An empty intersection makes the clue unsatisfiable. The power of this clue comes from parity reasoning: combine it with individual group totals to determine whether the overlap cell-set must contribute an odd or even number to each group's budget.
Reading the examples
“Y's neighbors and Z's neighbors share an odd number of suspects”
The clue says Alice's neighbors and Eve's neighbors share an odd number of suspects. Given the revealed cells, keeping the shared count odd forces Diana (Innocent).
“Y's neighbors and column A share an odd number of suspects”
The clue says Eve's neighbors and column A share an odd number of suspects. Given the revealed cells, keeping the shared count odd forces Grace (Innocent).
“Y's neighbors and row 1 share an odd number of suspects”
The clue says Eve's neighbors and row 1 share an odd number of suspects. Given the revealed cells, keeping the shared count odd forces Carol (Innocent).
“The edge and Y's neighbors share an odd number of suspects”
The clue says the edge and Eve's neighbors share an odd number of suspects. Given the revealed cells, keeping the shared count odd forces Iris (Innocent).
“The edge and row 1 share an odd number of suspects”
The clue says the edge and row 1 share an odd number of suspects. Given the revealed cells, keeping the shared count odd forces Carol (Innocent).
“Between Y and Z and W's neighbors share an odd number of suspects”
The clue says between Alice and Carol and Eve's neighbors share an odd number of suspects. Given the revealed cells, keeping the shared count odd forces Bob (Suspect).