Hidden Figures: Checking a Candidate's Fast Identification
You are coaching a candidate on Hidden Figures. On each item the candidate sees a jumble of labeled line segments drawn on a square grid and must decide which subset forms the stated target figure. In practice item 7 the target is a right isosceles triangle whose two legs are exactly 3 grid units long, axis-aligned, with the diagonal as the third side. The jumble contains: S1 (2,1)->(5,1); S2 (5,1)->(5,4); S3 (5,4)->(2,1); S4 (0,2)->(3,2); S5 (3,2)->(3,6); S6 (1,0)->(4,3); S7 (4,3)->(4,7); S8 (0,5)->(3,5); S9 (6,2)->(6,5); S10 (6,5)->(3,2). The candidate selects S1, S2, and S3 in about four seconds and moves on.
1. Does the candidate's selection satisfy every stated condition of the target figure? Show the checks.
Read the worked answer
Check each condition separately.
- Leg 1: S1 runs (2,1)->(5,1), purely horizontal, length 5 - 2 = 3 units.
- Leg 2: S2 runs (5,1)->(5,4), purely vertical, length 4 - 1 = 3 units.
- Right angle: S1 is horizontal and S2 is vertical, and they meet at (5,1), so the included angle is 90 degrees.
- Hypotenuse: the free ends are (2,1) and (5,4); S3 runs exactly (5,4)->(2,1), so it closes the triangle, and its length is sqrt(3^2 + 3^2) = 3*sqrt(2) ~ 4.24, the correct hypotenuse for legs of 3.
- No extra segments: the selection is exactly three segments.
The selection is correct. The four-second time is fast but the answer is not a guess; every condition is met.
2. A later item has 14 segments and the candidate has 60 seconds for 8 items. Should the candidate test subsets by trial and error? Quantify and propose a faster scan order.
Read the worked answer
Budget: 60 s / 8 items = 7.5 seconds per item.
Brute force: choosing triples from 14 segments gives C(14,3) = 364 possible subsets. Even at one second per check that is far beyond 7.5 seconds, so random triple-testing cannot work.
Anchor scan instead:
- Find corners - pairs of segments sharing an endpoint whose directions are perpendicular (horizontal vs. vertical). The target needs one of these, so this prunes the search drastically.
- At each corner, measure the two legs. If either differs from the required length, discard the corner immediately; do not go looking for a hypotenuse.
- Only after a corner passes the leg-length test, check whether the two far ends are joined by a single segment whose length equals the required hypotenuse.
Order matters: the corner test is the cheapest discriminator, and the hypotenuse check is the most expensive, so it goes last.
3. Explain why segments S9 and S10 are a plausible but wrong attractor.
Read the worked answer
S9 runs (6,2)->(6,5): vertical, length 3 - it matches the required leg length. S10 runs (6,5)->(3,2): length sqrt(3^2 + 3^2) = 3*sqrt(2), matching the required hypotenuse. The two also share the endpoint (6,5), so the length pattern looks like one leg plus a hypotenuse.
But the figure fails two independent tests:
- Angle: the angle at (6,5) between S9 (pointing straight down) and S10 (pointing down-left at 45 degrees) is 45 degrees, not 90.
- Closure: even treating S9 as a leg, the second leg would have to run from (6,2) to (3,2) horizontally. No such segment exists in the jumble.
S9 and S10 reward a candidate who pattern-matches on lengths alone. The leg-and-right-angle test kills them in about two seconds.