Spatial reasoning, oriented

A whole figure is hard to turn in your head, but one feature is easy. Choose the part that makes the shape lopsided, such as the short bar of a letter F, and ask only where it ends up. Then check the options for that feature in that place, and cross out every option where it is wrong. A second feature usually settles whatever is left.

Learn the turns as directions. A quarter turn clockwise sends a feature pointing up to point right, right to down, down to left and left to up, like a clock hand moving from 12 to 3. A half turn, 180°, sends every feature to the opposite side in both directions at once: top left ends at bottom right. If only one direction swapped, it was not a half turn. A 270° clockwise turn lands in the same place as a quarter turn anticlockwise, so treat it as the shorter turn the other way.

The trap is comparing whole figures and choosing the one that looks closest. Options are built to look close, and the eye settles too early. Tracking one feature replaces an impression with a check.

Rule: follow one feature, say its new direction aloud, and eliminate options that put it anywhere else.

Picture a mirror standing beside the figure. Whatever was on the left is now on the right, but nothing moves up or down. A letter F mirrored this way has its arms pointing left, while its upright keeps the same height. Check one feature for each direction: it should have swapped sides and kept its level.

Rotations keep a shape's handedness, the way a right glove stays a right glove however you turn it. A mirror swaps the handedness. So if no amount of turning makes a figure match the original, it is a mirror image. A practical test uses two features: under a rotation they keep the same arrangement relative to each other, and under a mirror that arrangement reverses.

A top-to-bottom flip, like a reflection in a lake, is the same as a left-to-right flip followed by a half turn. That lets you check an unfamiliar flip with two moves you can picture more easily.

Be careful with symmetric parts. A straight bar or a plain circle can look unchanged after a flip or a turn, so watching it tells you nothing. Track the part that breaks the symmetry, such as a foot on one side or a hook at one end.

Rule: a mirror swaps one direction and keeps the other, and no turn can ever undo it.

When an item applies two or three moves in order, do them one at a time on a single feature and note where it points after each step. Jumping straight to the end is where most errors come from, because two moves are hard to combine in your head.

Keep the order exactly as given. A flip followed by a quarter turn usually lands somewhere different from the same quarter turn followed by the flip. Only a half turn combines with a left-to-right flip the same way in either order. Wrong options are often the same moves done in another order, or the figure after only some of the moves.

Some chains shrink to a single move, which turns a long check into a short one. Two left-to-right flips cancel each other out. A flip, then a quarter turn clockwise, then another flip adds up to a quarter turn anticlockwise, because the flips reverse the direction of the turn between them.

If an option matches where your tracked feature was halfway through the chain, it is a trap built for a solver who stopped early. Finish every step before you choose.

Rule: one move at a time, in the given order, then simplify and check the final position only.

In a figure analogy, the first figure turns into the second by one move: a quarter turn, a half turn, a quarter turn the other way, a mirror flip, or a flip and a turn together. Name that move in words before you look at the third figure. The third figure is usually a different shape, so the answer will not resemble the second figure. What carries over is the move. Track one feature of the third figure through it and find the option that matches.

A turning sequence hides a constant step, usually a quarter turn in one direction. Find it between the first two figures, confirm it between the second and third, and apply it once more. Four quarter turns make a full circle, so the fifth figure in such a sequence matches the first, which helps when a sequence is long.

In an odd-one-out set, three figures share an orientation and one does not. Check a single feature across all four, such as which way an arm points, and the exception shows itself. You do not need to work out how it was moved. If no turn would make the exception match the others, it has been flipped, and it is still the answer.

The common error in all three is matching appearance instead of naming the change. Words such as "quarter turn clockwise" turn an impression into something you can apply.

Rule: name the move in words, then apply it, continue it, or find the figure that does not follow it.

In a scattered figure the eye jumps around and loses its place. Sweep it the way you read a page, left to right along each row and then down to the next. A steady route passes each shape exactly once, which removes both the missed shape and the double count. Count only the shape asked for, keeping a running tally. Then, if you have time, count the other shapes too and check that the totals add up to the whole figure. The wrong options sit one or two away from the true count, so a single slip is enough to land on one.

Questions that say "of any size" want every square or rectangle the grid lines form. For squares, count each size separately: in a grid w cells wide and h tall, a k × k square fits (w − k + 1) × (h − k + 1) ways. A 2 × 8 grid has 16 one-by-one squares and 7 two-by-two squares, 23 in all.

For rectangles, use the gridlines. Every rectangle is fixed by choosing two vertical lines and two horizontal lines. A grid 2 cells wide and 6 tall has 3 vertical and 7 horizontal lines, so there are 3 × 21 = 63 rectangles. If your count equals the number of small cells, you have answered a different question.

Rule: count in a fixed order, and for any size, count by size or by gridlines, never by cells.