The Museum Arrow, Frame Six
A museum installation shows a single arrow on a square panel, changing once every two seconds, and visitors are invited to predict frame 6 before it appears. Frames 1 to 5 have been recorded. Frame 1 is a long unfilled arrow pointing north. Frame 2 is a short filled arrow pointing north-east. Frame 3 is a long unfilled arrow pointing east. Frame 4 is a short filled arrow pointing south-east. Frame 5 is a long unfilled arrow pointing south. The panel can draw arrows at the eight compass directions only, in two lengths (long and short) and two finishes (unfilled and filled); no other frame is possible. One visitor forecasts a long arrow pointing west, on the grounds that the arrow turns 90 degrees clockwise each time. A second visitor forecasts a short arrow pointing south-west. The curator needs one forecast for the exhibit label, plus a written justification visitors can follow.
1. Which forecast is correct? Give the full rule and explain why the 90-degree forecast fails.
Read the worked answer
The second visitor is right: frame 6 is a short filled arrow pointing south-west.
Three attributes move together, one step per frame:
- Direction turns 45 degrees clockwise: north, north-east, east, south-east, south, so frame 6 is south-west.
- Length alternates long, short, long, short, long, so frame 6 is short.
- Finish alternates unfilled, filled, unfilled, filled, unfilled, so frame 6 is filled.
Each rule is confirmed four times over the five recorded frames, and the three never collide.
The 90-degree forecast comes from reading only frames 1, 3 and 5, which sit at north, east and south. That reading leaves frames 2 and 4 completely unexplained - they sit at north-east and south-east, exactly halfway between the odd frames, which is impossible if the arrow jumps a full quarter turn each step. A rule that ignores half the evidence is not a simpler rule, it is an incomplete one.
2. Frames 1, 3 and 5 alone are consistent with a 90-degree turn. Explain what frames 2 and 4 contribute, and state the general principle a solver should apply to serial patterns.
Read the worked answer
The odd frames cannot settle the step size, because any three points can be fitted by more than one constant turn. North to east to south is consistent with 90 degrees per frame and also with 45 degrees per frame if you simply skip a frame each time.
Frames 2 and 4 break the tie. They land at north-east and south-east, precisely the positions a 45-degree turn predicts and precisely the positions a 90-degree turn cannot produce. Once the even frames are included, only one constant step survives.
The principle: test every frame you have, not the convenient subset. Serial patterns are usually built so that the 'obvious' reading works on the frames you look at first and fails on the ones you skipped. Also, a step is not established by a single interval - it needs at least two repetitions, plus the absence of any frame that contradicts it. Here the 45-degree rule passes both tests and the 90-degree rule fails the second.
3. Predict frames 7 and 8, state at which frame the sequence first repeats frame 1, and explain how the panel's total set of possible frames affects how confident you can be.
Read the worked answer
Frame 7 is a long unfilled arrow pointing west. Frame 8 is a short filled arrow pointing north-west.
Frame 9 returns to a long unfilled arrow pointing north, which is frame 1 exactly. Direction has a period of eight (the eight compass points); length and finish have a period of two. The combined period is the least common multiple, eight, so the animation loops every eight frames, or every 16 seconds.
The panel can display 8 directions x 2 lengths x 2 finishes = 32 distinct frames, but only eight are ever used, because length and finish are locked to the direction's parity: every even-numbered direction is short and filled, every odd-numbered one long and unfilled. That is worth knowing when judging confidence: with 32 states and only five observations, many rules could fit a subset, so the forecast rests not on the number of possibilities but on the single rule that explains all five recorded frames without exception.