Stencil Pair on the Sign-Shop Bench
A sign shop uses pairs of stencilled plates. Plate A shows a small open circle with a small filled triangle attached at its right-hand side. Plate B shows a large open circle with the same filled triangle attached above it. The workshop's standing rule is that B is produced from A by exactly two changes applied together, and the same two changes must turn plate C into plate D. Plate C shows a small open square with a small filled star attached at its left-hand side. Four candidate plates for D are on the bench: (i) a large open square with the star above it; (ii) a large open square with the star below it; (iii) a small open square with the star below it; (iv) a large open square with the star to the right. An apprentice has already cut plate (ii) and wants to ship it, but the foreman insists the rule be written down first. The order leaves with exactly one D plate, and any plate cut in error is scrap.
1. Which candidate is plate D, and what are the two changes? Show how each of the other three candidates fails.
Read the worked answer
Plate D is (ii), a large open square with the star below it.
The two changes are: the main shape grows from small to large, and the satellite swings a quarter turn anticlockwise about the main shape (right becomes top).
Applying them to C: the square grows, ruling out (iii), which stays small. The star, which sits on the left, swings a quarter turn anticlockwise from left to bottom.
Failure of the others:
- (i) star above keeps the size but treats the move as a mirror image across a horizontal line - the star has been reflected, not rotated.
- (iii) star below, small gets the direction right and the size wrong, so only one of the two changes was applied.
- (iv) star to the right applies the quarter turn clockwise, reversing the rotation; left would have to become top, not right, under the reversed rule.
Only (ii) reproduces both changes at once.
2. A colleague argues the A-to-B move could equally be described as reflecting the satellite across the diagonal running from bottom-left to top-right, rather than rotating it. Show that both descriptions give the same D here, and say what that means for confidence in the answer.
Read the worked answer
Both descriptions do fit the A/B pair. A quarter turn anticlockwise sends a right-hand attachment to the top. A reflection across the bottom-left-to-top-right diagonal also sends the right-hand position to the top. With a single example pair, the two rules are indistinguishable.
That matters because of where C puts its star. A quarter turn anticlockwise sends the left-hand position to the bottom. Reflecting the left-hand point across that same diagonal also sends it to the bottom. So both readings select plate (ii), and the answer survives the ambiguity intact.
The general habit: when two rules both fit the worked example, do not agonise over which is 'really' intended - apply both to the target and see whether they diverge. If they agree, the item has one defensible answer and can be marked. If they diverge, the item is flawed and no amount of re-reading will fix it.
3. What would happen to the item if C's star had been attached at the top of the square instead of the left? Explain, and state the general consequence for analogy items.
Read the worked answer
The two rules would part company and the item would have two defensible answers.
If the star sits on top of C: a quarter turn anticlockwise sends it to the left, giving a large square with the star on the left. A reflection across the bottom-left-to-top-right diagonal sends the top position to the right, giving a large square with the star on the right. Both plates would be equally consistent with the A/B pair, so neither could be justified over the other, and the problem would be unanswerable rather than hard.
So the choice of 'star on the left' in C is not decoration; it is the design feature that makes exactly one answer defensible. For a solver, the practical test is: find a second rule that fits the example, run it on the target, and only then commit. If the second rule gives a different target, you need a second example pair to break the tie before answering.