6 models P to U of a particular product are lined for inspection. They are not…

Concept: In a single-row arrangement with empty locations, first lock every position forced directly by a clue (a named location, an end-of-row limit, or a "no other option" deduction). Then use the count clue ("exactly k models are next to an empty location") to test which empty-location pattern(s) survive: an interior block of adjacent empty locations borders exactly two occupied locations, while a block touching either end of the row, or two separated empty locations, borders a different number. When more than one empty-location pattern survives every clue, a location is "certain" only if it holds the SAME value -- a named model or "empty" -- in every surviving pattern.

Application: Trace the clues for this arrangement of 8 locations (1 to 8).

1. T is fixed at location 7 (clue A).

2. Clue B needs a model on BOTH sides of T. Location 8 is the last location in the row (there is no location 9), so if location 8 were empty, T would have no model at all on that side. Hence location 8 must be occupied -- by Q or R.

3. Clue B only fixes WHICH two models (Q and R) are nearest to T, not that they sit immediately beside it. So the model nearest on the left could be at location 6 (if location 6 is occupied) or further left, if location 6 (or locations 5 and 6) are empty.

4. Clue E fixes exactly two empty locations; clue D removes location 3; locations 7 and 8 are already occupied. So the two empty locations must be chosen from {1, 2, 4, 5, 6}.

5. Testing every 2-location choice from {1, 2, 4, 5, 6} against clue C (exactly two models sit next to an empty location): the adjacent end pair {1, 2} borders only ONE occupied location (location 3), because there is no location 0 -- that gives a count of 1, not 2, so it fails. Any pair of empty locations that are not adjacent to each other (including any other pair that contains location 1) borders three or four occupied locations -- that also fails. Only two choices survive the count-of-two test: {4, 5} and {5, 6}.

6. Case 1 -- empty locations {4, 5}: location 6 is occupied, so it is the immediate left neighbour of T and holds Q or R; P, S and U fill locations 1, 2, 3 in some order.

7. Case 2 -- empty locations {5, 6}: location 6 is empty, so the nearest occupied location to T on the left is location 4, which holds Q or R; P, S and U again fill locations 1, 2, 3 in some order.

8. Both cases satisfy clues A to E, so both remain valid -- the arrangement is genuinely case-split at this point.

Cross-check: Compare every location across the two surviving cases.

1. Location 5 -- empty in both cases -> certain (empty).

2. Location 7 -- holds T in both cases -> certain (T).

3. Location 4 -- empty in case 1 but occupied by Q or R in case 2 -> not certain.

4. Location 6 -- occupied by Q or R in case 1 but empty in case 2 -> not certain.

5. Locations 1, 2, 3 and 8 -- occupied in both cases, but no clue ever pins down WHICH model (P, S or U at 1-3; Q or R at 8) sits there -> not certain.

So exactly two locations -- location 5 (empty) and location 7 (T) -- keep the same status in every layout consistent with the clues, giving the answer 2.