A Sequence That Repeats Every Four Frames
Marcus works a serial reasoning item: five frames are shown and he must choose the sixth. Each frame is a square with a small pennant attached to one edge, plus one dot inside the square. Across the five frames the pennant travels north, east, south, west, north. The dot sits near the top edge in frames 1, 3 and 5, and near the bottom edge in frames 2 and 4.
The options are A pennant east, dot at top; B pennant east, dot at bottom; C pennant north, dot at bottom; D pennant south, dot at top; E pennant west, dot at bottom.
Marcus marks A. He says: "Frame 5 is the same as frame 1, which I think is a printing mistake, so I started over from frame 1 and rotated one step. The dot was confusing me, so I ignored it - it looked like it just resets." He finishes the section with three items unmarked, all of them items where the first and last shown frames were identical.
1. Which option is correct, and how does the apparent repetition in frame 5 confirm rather than undermine it?
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Answer: B. The frame has two independent cycles:
- Pennant: north, east, south, west - a 90-degree clockwise turn each step, period 4. Frame 5 returns to north, exactly as the cycle predicts.
- Dot: top, bottom, top, bottom - a 180-degree move each step, period 2.
A frame is the combination of both, so the whole pattern repeats at the least common multiple of 4 and 2, which is 4. Frame 5 should equal frame 1, and it does. That is the check that the reading is right, not a printing error.
Therefore frame 6 equals frame 2: pennant east, dot at bottom. Marcus got the pennant right and threw away the dot. Note that on frame 5 the dot also returns to the top, which is precisely what tempted him to call it a reset.
2. Explain Marcus's error in a way that names the general failure, and connect it to the items he left blank.
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Marcus treated a two-channel sequence as a one-channel sequence. He correctly decoded the pennant and then discarded the dot as noise because the dot's behaviour did not match the pattern he had already committed to. That is not a careless slip; it is a filtering decision made before the second rule was tested.
The tell is that he had the information to catch himself. He noticed frame 5 equals frame 1 and felt uneasy - but instead of asking "does my rule predict this?", he assumed the item was faulty. When a prediction and a page disagree, the safer default is that the rule is incomplete.
This explains his unmarked items. Where the first and last shown frames match, the pattern's period divides the number of frames shown, and every channel looks like it has reset. Those items reward exactly the reasoning he abandons. His practical fix: before choosing, list every element that varies - here two - and state a rule for each.
3. What does frame 23 look like, and what is the shortest route to that answer?
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Frame 23 has the pennant south and the dot at the top - identical to frame 3.
The short route is to stop stepping and use the period. The configuration repeats every 4 frames, so frame n matches frame r where r is the remainder when n is divided by 4, with remainder 0 meaning frame 4. For 23: 23 = 4 × 5 + 3, so the remainder is 3, and frame 23 matches frame 3.
Check it against each channel separately, which catches arithmetic slips. The pennant advances 90 degrees per frame; 22 steps after frame 1 is 22 × 90 = 1980 degrees, and 1980 divided by 360 leaves 180, so the pennant ends opposite north, which is south. The dot alternates; frame 1 is odd and top, frame 23 is odd and top. Both channels agree with frame 3.
Stepping frame by frame to 23 is possible but slow and error-prone; deriving the period and using a remainder takes a few seconds.