Assembling Objects: mixed review

The page frame is fixed (up, down, left, right of the sheet). The figure frame travels with the shape (its own top, front, sides). State which frame you mean; mixing them causes most errors.

90° CCW = 270° CW. A 180° turn is the same either way. Add CW turns as positive, CCW as negative, and drop full 360° circles: 90 + 180 − 90 = a half-turn.

$(r,c)\to (c,n+1-r)$, rows counted from the top. On a 4 by 4 grid, (2, 1) moves to (1, 3).

CCW: $(r,c)\to (n+1-c,r)$. Half-turn: $(r,c)\to (n+1-r,n+1-c)$. On a 3 by 3 grid, CCW sends (1, 1) to (3, 1).

Read three landmarks clockwise. Rotations and slides keep that cyclic order (any starting point); a reflection reverses it. Red-blue-green becoming blue-green-red is a rotation; red-green-blue is a mirror.

A half-turn: (x, y) becomes (−x, −y). The two mirror effects cancel, so the result is a true rotation, not a mirror image.

Different motions can send one cell to the same place: on a 3 by 3 grid both a left-right flip and a clockwise quarter-turn move (1, 1) to (1, 3). Use a second landmark.

Add the piece areas and compare with the finished outline. Pieces of 6, 4 and 2 square units (12) can fill a 3 by 4 rectangle but never a 3 by 5 (15). Any mismatch rejects the choice.

Opposite directions: north pairs with south, east with west. If the fixed piece's port faces east, the partner must present a west-facing edge.

Clockwise 90°: N to E, E to S, S to W, W to N. Counterclockwise reverses this. 180° swaps opposites (N with S, E with W).

It moves with the exact spot it marks. A dot one third of the way along the long edge stays one third along that edge; decoys slide it to the middle or a neighboring edge.

The distance between the two marks must be equal on both pieces, $d_1(K,R)=d_2(K,R)$, because rigid pieces keep their spacing. Marks 4 units apart cannot match marks 6 units apart.

One shared point fixes location but lets the piece spin around it. Orientation needs a second point, a required facing edge or a connector shape.

They reverse. If piece 1 reads A then B clockwise, the partner edge must read B then A clockwise. Same order on both means a flip would be needed.

Tabs pair with recesses (out with in, flat with flat). The slot must be as wide as the tab and as deep as the tab is long; too narrow jams, too shallow leaves a gap.

Identity (right partners), position (same edge and spot on the piece), orientation (a rotation, not a mirror, same size), fit (connectors complement, zero overlap). Reject at the first failure.

Clockwise (12 toward 3): top goes to the right. Counterclockwise (12 toward 9): top goes to the left. 90° CCW equals 270° CW, and a half-turn is the same in either direction.

Clockwise: $(r,c)\to (c,6-r)$. Counterclockwise: $(r,c)\to (6-c,r)$. Rows count from the top, columns from the left; (1, 4) turns clockwise to (4, 5).

$(r,c)\to (6-r,6-c)$. Cell (1, 4) goes to (5, 2), the center (3, 3) stays put, and doing it twice returns every cell home.

90° CCW: $(x,y)\to (-y,x)$. 90° CW: $(x,y)\to (y,-x)$. 180°: $(x,y)\to (-x,-y)$. Example: (3, 1) goes to (−1, 3), (1, −3) and (−3, −1).

Row numbers grow downward while y grows upward, so the sense of the turn reverses: that rule gives a clockwise picture. Use the row-column rules and test a corner cell.

Whether a shape is a left- or right-handed arrangement, like your two hands. Rotations never change it; a flip reverses it. Spinning b gives q (half-turn); b and d are mirror images.

1) Count: same squares, arm lengths and connections. 2) Direction: turn an imaginary arrow and drop wrong facings. 3) Handedness: check a side-specific feature to drop the mirror copy.

No. A marker is welded to its cell and turns with the shape, staying in the same part of it (for example the end of the short arm). Check outline and marker together.

Exactly the sum of the pieces' cell counts (7 + 5 = 12). Fewer cells means an overlap or lost material; more means invented material.

Edge contact: pieces share a full side of at least one cell. Corner contact: they touch at a single point, like diagonal checkerboard squares. A required marked-edge join needs flush edge contact.

Exactly once, with its shape unchanged. A missing piece leaves the whole incomplete; a duplicate adds material that was never supplied, even if the outline looks tidy.

Not necessarily. Different arrangements can share one outline while seams and connector marks sit in different places. After the outline passes, check seams and marks.

A right-side dot pairs with a left-side dot in the cell directly to its right: same row, columns differ by exactly 1. Top-to-bottom joints share a column with rows 1 apart.

It barely changes the picture but moves the mark a full cell: a horizontal shift leaves a gap, a vertical shift puts the marks on different rows. Almost correct is wrong.

List features clockwise around the edge (long arm, notch, short arm). A turn keeps that cyclic order and only changes the starting point; a mirror image reverses it.

First check piece count and total cells, then that each piece keeps its shape with no flip, then that every pair of marks touches exactly. A correct cell count alone never proves orientation.

A closer opaque surface blocking the view of part of an object behind it. The rear object is unchanged; a missing line in the picture does not mean a missing piece.

Closer to the observer along the viewing direction, not higher on the page. Use the viewing arrow or stated layer order; viewing from the opposite side can reverse which part hides the other.

Imagine a straight ray from the eye to the point. If it hits a closer opaque surface first, the point is hidden; if nothing closer blocks it, the point is visible.

No. Occlusion hides, it does not remove material. A straight line stays straight, a diagonal's two exposed tails keep the same slope, a curve stays smooth and a hidden tip stays pointed.

No. Dashed hidden lines are a drafting convention for showing concealed edges. An ordinary visible-surface view shows nothing of the rear object inside the cover unless hidden lines are requested.

With occlusion the exposed rear stroke ends exactly at the cover's boundary. A stroke that stops short of the cover leaves extra blank space and is wrong. Check both entry and exit points.

The actual cover outline where the line crosses it. A circle hides a longer span through its center than near its top; a diamond's sloped edges stop the line, not its bounding rectangle.

Reject in order: wrong cover shape or position, rear line drawn over the cover, a missing exposed tail or arm, then subtle entry and exit mismatches against the source shapes.

A feature that must stay constant under the allowed motion. Under translation, shape, cell arrangement and protrusion direction all stay fixed; under rotation, distances stay fixed but directions change.

Exactly when $x_Y=x_X+1$ and $y_Y=y_X$: Y is in the next column on the same row. Rows one apart mean a slip; columns two apart mean a gap.

No. It is a necessary screen only (6 + 5 cells must show 11), but different 11-cell layouts exist. Still check shape, orientation and the exact connector placement.

Face contact shares a full unit boundary between cells of the two pieces. Diagonal corner contact touches at a single point and does not count as a join.

Pick two distinct landmarks (a top notch and a lower spur) and note their left-right order, or trace landmarks clockwise. A mirrored candidate reverses that order even with the same cell count.

No. Extra contact is fine unless the prompt forbids it. Reject only for a stated rule: the marked X and Y must meet. Never add rules the prompt does not give.

Pass 1: quickly drop clear violations (extra or missing piece, forbidden mirror, open gap). Pass 2: on the survivors, compare marked cells and one distinctive landmark, then positively verify the final choice.

No. The answer must satisfy every stated condition at once, judged as printed. A tidy silhouette that needs one edit (shift, unflip) is still a distractor.

1) What moves are allowed? 2) Where is the anchor or marked point? 3) What is the unusual contour, notch or protrusion? 4) Which edges must touch? 5) Which candidate breaks an invariant?

Rotate just one distinctive arm or marker once to predict where it should point, then scan each option for that prediction. Rotate the full shape only after narrowing the field.

As a secondary check, when a distractor may add or drop a square or outlines nearly match. Use the cheapest decisive check first: an arm's direction takes one glance, counting takes longer.

Start with whatever best separates the choices: if outlines are similar but dot positions differ, check the marker first; if outlines differ strongly, check the silhouette first.

No. Once submitted, an answer cannot be revisited, so use a scan, compare, verify cycle before submitting rather than planning to flag and return.

No. Official guidance warns that a string of wrong answers near the end can bring a penalty. Keep eliminating visible errors and pick the best-supported choice.

Reusing the previous item's angle or swapping clockwise and counterclockwise. Reread the instruction and name it first: clockwise sends the top toward the right.

Pass 1 rejects quickly with one invariant (for example a forbidden mirror). Pass 2 checks survivors with a different invariant (marker position or cell count). Still verify the last option.

A slide keeps facing and handedness. A rotation changes facing but keeps handedness. A reflection reverses handedness, so it is illegal unless the prompt allows it.

Number of filled cells, lengths of each rigid side, the order of features around the piece, and which cell owns each marked point. All survive any legal slide or turn.

Fix piece A, move only B relative to it and secure that contact. For a third piece, treat the A–B union as the new anchor and add C the same way.

Turn the piece about its own marked point, then slide that mark to its target, then check for collision. If a global rotation center is stated, use that center.

Pass A: reject rule breakers (forbidden mirror, wrong turn, wrong contact). Pass B: test invariants (cells, mark identity, notch). Pass C: verify edge contact, no overlap, correct silhouette.

Mirror (handedness reversed), rotation (wrong angle), placement (right shape, broken contact), marker (wrong marked cell used) and conservation (material added or removed).

15 scored questions in 18 minutes when no tryout items are included (more time with tryouts). Answers cannot be changed after submission; paper testing can allow review within a subtest.

Only the exact stated operations and contacts count. A neat fit from a different turn, or a blue-to-red contact when red must touch red, is still wrong.