Hard Items: A Narrow World

The whole set fits inside nine degrees

Here is a hard row. Before reading on, try to rank the four openings from smallest to largest.

You almost certainly could not, and the reason is not that you were careless. On a hard item the three gaps between neighboring angles are two or three degrees each, so the set from smallest to largest spans only six to nine degrees in total. Four angles inside a window that narrow do not read as four sizes. They read as one angle drawn four times.

One hundred nineteen to one hundred twenty-seven — eight degrees end to end, and two of the four are two degrees apart. That is the world a hard item lives in, and no amount of care changes its width.

Only the separation shrank

Now the same four panels again. Identical orientations, identical ray lengths, identical layout — the only thing changed is that the angles have been pushed apart into the easy band.

Suddenly the ranking is obvious, and nothing about the drawing helped you. Difficulty in this section is not complexity, it is margin. The panels never get busier and no new kind of figure shows up; the four angles simply move closer together, and everything that made you accurate at easy has less to work with. That is worth knowing before test day, because it tells you what a hard item is not. It is not a puzzle with a trick in it, and there is nothing extra to find.

Coarse sorting stops paying

The cheapest thing you can do to a set of angles is split it at a right angle: these are clearly under ninety degrees, those are clearly over. When a set straddles ninety, the split is a free partial ordering — every acute angle outranks every obtuse one and you have not made a single fine judgment.

It stops working almost entirely on hard items. Measured over four thousand hard items from this course's generator, 94 percent had all four angles on the same side of ninety degrees; at easy the figure is 64 percent. Put the other way round: an easy set straddles the right angle about one time in three, a hard set about one time in sixteen. The row above is typical — all four obtuse, so the split returns all four in one bucket and eliminates nothing.

The arithmetic is not mysterious: a set only straddles ninety if the boundary happens to fall inside its span, and a hard span is eight degrees where an easy one is nearer forty. Narrow sets miss the boundary. So do not spend the first look on a hard item sorting by right angles — the answer is almost always "all of them, same side".

The noise did not shrink with the signal

Ray lengths and orientations are drawn from the same ranges at every difficulty. One angle can still carry rays half again as long as its neighbor's, and any angle can still be turned to any heading. None of that intensifies on hard items — but its relative damage does, because the real difference between two angles has shrunk to two degrees while the distraction has not shrunk at all.

Look back at the hard row. Panel #1 carries the longest rays in the row and, by some margin, the widest tip-to-tip gap of the four — and it is the second smallest opening. At easy a mismatch like that gets overruled by a thirteen-degree difference. At hard there is no thirteen-degree difference anywhere in the item to overrule it.

The measurement matches the impression: rank the four panels by how far apart their ray tips end up — a mechanical stand-in for the untrained glance — and that ranking disagrees with the truth on about 42 percent of individual pairs at hard, against 22 percent at easy. Two pairs in five is not a hint you can lean on.

What survives

Three things, and none of them are new techniques.

• The extremes are still the widest comparison you have. Smallest against largest spans every gap in the set at once — six to nine degrees on a hard item, which is about as wide as the narrowest gap an easy item ever draws. It is the only comparison in a hard item with any margin at all, so it is the one to make first and the one to trust.
• The middle pair goes last, because it is separated by a single gap and on a hard item that gap is two or three degrees. Deciding it early contaminates the judgments that were actually sound.
• An answer submitted beats an answer perfected. There is no reading of a two-degree gap that a fourth look will improve.

That is the hard row's two extremes, lifted out and set side by side. Eight degrees, judged against each other with nothing else in the way, is a call most people can make — which is the entire argument for making it first.

Recognizing one in two seconds

You do not need to be told a set is hard. The tell is immediate: you scan the row and no panel volunteers as the smallest. On an easy set one opening is obviously the runt and another obviously the fattest, and you know it before you have decided to look. When nobody volunteers, the set is narrow.

Switch right there, on that first scan. Stop trying to take in the whole ranking at once — on a set this tight that produces an impression, and the impression is about as reliable as the ray tips. Go straight to the two extremes and work from the only margin the item gave you.

The takeaway

A hard item is four angles inside a six-to-nine degree window — the same drawing, the same ray-length spread, the same orientations, with the margin taken out. Splitting the set at ninety degrees stops helping, because 94 percent of hard sets sit entirely on one side of it. The ray-length noise is unchanged in size and therefore much larger relative to the signal, which is why an impression of the whole row is worth so little. The extremes still span every gap at once, so they remain the one comparison worth making first, and the middle pair remains the one to leave until last.