Circular Arrangement with a Mirror Solution
A mock test for PGT General Intelligence contains a circular-arrangement item. Six teachers P, Q, R, S, T and U sit around a table facing the centre. The clues are: (1) P sits opposite S. (2) Q sits opposite T. (3) R sits opposite U. (4) Q sits adjacent to P and R. (5) U sits adjacent to T and P. The question asks: Who sits second to the right of P? The key says T. During review, a subject expert produces a second clockwise order - P, U, T, S, R, Q - and claims it satisfies every clue but gives R as the answer. The item setter argues the second order is just a rotation and should be ignored. The review committee has to decide whether the item can be retained, and if not, what minimal change makes it single-answer. The test has 100 items and 4,000 candidates; a flawed key would affect ranks.
1. Test both clockwise orders against all five clues. Does the second order genuinely satisfy them, and why does the setter's rotation argument fail?
Read the worked answer
Check Arrangement I clockwise as P, Q, R, S, T, U. P is opposite S (positions 1 and 4); Q opposite T (2 and 5); R opposite U (3 and 6); Q is adjacent to P and R; U is adjacent to T and P. All five clues hold.
Check Arrangement II clockwise as P, U, T, S, R, Q. P opposite S (1 and 4); Q opposite T (6 and 3); R opposite U (5 and 2); Q is adjacent to P and R (6 touches 1 and 5); U is adjacent to T and P (2 touches 3 and 1). All five clues hold.
The setter's rotation argument fails because Arrangement II is not a rotation of Arrangement I. A rotation would preserve the clockwise sequence P-Q-R-S-T-U; Arrangement II reverses it to P-U-T-S-R-Q. Reversal preserves adjacency and oppositeness but swaps left and right. Because every clue is symmetric, the clues cannot choose between the two. The directional question therefore has two answers: T in Arrangement I and R in Arrangement II under the standard convention that a person facing the centre has their right anticlockwise.
2. Add one minimal clue that fixes the orientation and state the answer to the original question under that clue.
Read the worked answer
Add the clue: Q sits immediately to the left of P. For a person facing the centre, left is clockwise, so the clockwise neighbour of P must be Q. Arrangement I satisfies this (P then Q clockwise); Arrangement II does not (P then U clockwise). The added clue fixes the orientation, leaving Arrangement I and the original answer T. A minimal alternative is 'U sits immediately to the right of P', which selects Arrangement II and makes the answer R. The important point is that the added clue must be directional, not another adjacency or opposite clue.
3. What item-writing principle does this illustrate? Recommend a checklist for circular arrangement items.
Read the worked answer
Principle: a circular arrangement item that asks a left/right question must contain at least one clue that fixes handedness. Symmetric clues can only determine the seating up to reflection. A practical checklist: (1) specify facing centre or facing outward; (2) include one unambiguous left/right or clock-position clue; (3) test for a mirror solution by reversing the order; (4) avoid redundant clues that create an illusion of uniqueness; (5) have two solvers derive the answer independently; (6) if the item is already in the paper, drop it rather than defend a key that depends on an unstated convention.