This 39-question test checks the core skills from Spatial Orientation Basics. The original questions ask you to follow a figure through a quarter-turn or half-turn, tell a rotation from a reflection, track a marked corner, map a tile on a grid, and reject candidates that are mirrored, altered or missing material.
A hiker stands facing north and makes one quarter-turn clockwise. Which direction is the hiker facing now?
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Answer: D
Correct answer: East. Going clockwise, the compass order is north, east, south, west, and each step is a 90-degree turn. One step clockwise from north is east.
A: North would mean no net turn, or a full 360-degree turn, not a quarter-turn.
B: West comes from turning a quarter-turn counterclockwise, which is the opposite direction.
C: South would need two quarter-turns, a half-turn, not one.
D (correct): Clockwise from north the order is north, east, south, west, so one quarter-turn clockwise lands on east.
Question 2 of 39 · easy
Which turn gives exactly the same result whether it is made clockwise or counterclockwise?
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Answer: D
Correct answer: A 180-degree turn. Turning 180 degrees clockwise and 180 degrees counterclockwise both reach the halfway point of the full 360-degree circle. That is why the direction of a half-turn never matters.
A: A quarter-turn clockwise points an up arrow right, while a quarter-turn counterclockwise points it left, so direction matters.
B: A 270-degree clockwise turn equals a 90-degree counterclockwise turn, so the two directions give different results.
C: A 45-degree turn one way tilts the shape opposite to a 45-degree turn the other way, so direction matters.
D (correct): A half-turn ends in the same place either way, since both directions swap top with bottom and left with right.
Question 3 of 39 · easy
On a 3 by 3 grid, rows are numbered from the top and columns from the left, and a cell is written (r, c). Which cell is the bottom-left corner?
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Answer: C
Correct answer: (3, 1). The bottom row is row 3 because rows count down from the top, and the left column is column 1. So the bottom-left corner is (3, 1).
A: This is the bottom row but the last column, the bottom-right corner.
B: This is row 1, column 3, the top-right corner.
C (correct): Row 3 is the bottom row and column 1 is the leftmost column, so this is the bottom-left corner.
D: This is row 1, column 1, the top-left corner.
Question 4 of 39 · easy
Which property of a shape can a reflection change, even though a rotation never changes it?
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Answer: C
Correct answer: The clockwise order of its landmarks. Rotations, translations and reflections all keep size, shape and area. The one thing a reflection changes is handedness: reading three landmarks clockwise gives the reversed order after a flip.
A: Every rigid motion, reflections included, keeps the size of each interior angle.
B: Every rigid motion, including a reflection, keeps the area the same.
C (correct): A reflection reverses the clockwise order of landmarks (handedness), while turns and shifts keep it.
D: Reflections are rigid motions, so they keep every side length.
Question 5 of 39 · easy
A piece is turned 90 degrees counterclockwise. The same final position can be reached by a clockwise turn of how many degrees?
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Answer: D
Correct answer: 270 degrees. Turning the other way around the circle covers the leftover amount:
360 - 90 = 270
So 90 degrees counterclockwise is the same as 270 degrees clockwise.
A: A 90-degree clockwise turn goes the opposite way and gives a different result.
B: A full turn returns the piece to its starting position, so it is not a 90-degree turn at all.
C: A half-turn is two quarter-turns, which overshoots by one quarter-turn.
D (correct): A full circle is 360 degrees, so 90 degrees counterclockwise matches 360 - 90 = 270 degrees clockwise.
Question 6 of 39 · easy
A shape is moved 4 units to the right and 2 units up without turning or flipping. In ordinary (x, y) coordinates, where does its point (1, 3) end up?
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Answer: B
Correct answer: (5, 5). A translation adds the same amounts to every point: x increases by 4 and y increases by 2. So (1, 3) moves to (5, 5).
A: This subtracts 4 from x, which moves the point left instead of right.
B (correct): Adding 4 to x and 2 to y gives (1 + 4, 3 + 2) = (5, 5).
C: This moves the point 2 units down instead of up.
D: This adds the 2 to x and the 4 to y, mixing up the two amounts.
Question 7 of 39 · easy
A scout stands facing east. Which compass direction is on the scout's left-hand side?
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Answer: D
Correct answer: North. Your left side is a quarter-turn counterclockwise from the way you face. Counterclockwise from east is north, so north is on the left.
A: East is the way the scout faces, straight ahead, not to the left.
B: South is on the right-hand side when facing east.
C: West is directly behind someone facing east.
D (correct): Facing east, the left hand points one quarter-turn counterclockwise from east, which is north.
Question 8 of 39 · easy
A plain shape with no markings is turned 120 degrees about its center. Which shape looks exactly the same as it did before the turn?
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Answer: B
Correct answer: An equilateral triangle. A shape with three equal sides and corners repeats its outline every
360 / 3 = 120 degrees
So an equilateral triangle looks the same after a 120-degree turn.
A: A square matches itself only after turns of 90, 180 or 270 degrees, and 120 is not one of them.
B (correct): Its three equal sides and corners repeat every 360 / 3 = 120 degrees, so a 120-degree turn looks unchanged.
C: A non-square rectangle matches itself only after a half-turn.
D: A regular pentagon repeats every 72 degrees, and 120 is not a multiple of 72.
Question 9 of 39 · easy
A square tile has a dot on its top edge. The tile is flipped over a horizontal mirror line. On which edge is the dot now?
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Answer: C
Correct answer: The bottom edge. A horizontal mirror line swaps top and bottom while leaving left and right alone. The dot on the top edge therefore ends up on the bottom edge.
A: The dot would stay on top only for a vertical mirror line, which swaps left and right instead.
B: A dot moves from the top to the left after a quarter-turn counterclockwise, not after a flip.
C (correct): A horizontal mirror line swaps top and bottom, so a dot on the top edge moves to the bottom edge.
D: A dot moves from the top to the right after a quarter-turn clockwise, not after a flip.
Question 10 of 39 · easy
On a flat piece, what is a vertex?
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Answer: D
Correct answer: A point where two edges meet. An edge is a straight side, and a vertex is the corner where two edges meet. A connection point at a vertex is different from one in the middle of an edge.
A: That describes an edge, not a vertex.
B: That describes an interior seam, not a vertex.
C: A midpoint lies along an edge, between its two ends, so it is not where edges meet.
D (correct): A vertex, also called a corner, is the spot where two edges come together.
Question 11 of 39 · medium
A piece is turned 90 degrees clockwise, then 270 degrees clockwise, then 90 degrees counterclockwise. What single turn has the same effect?
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Answer: B
Correct answer: A quarter-turn counterclockwise. Count clockwise as positive and counterclockwise as negative:
90+270−90=270
A 270-degree clockwise turn equals
360 - 270 = 90 degrees
counterclockwise.
A: This comes from adding all three turns as clockwise: 90 + 270 + 90 = 450, or 90 after removing a full circle.
B (correct): The net is 90 + 270 - 90 = 270 degrees clockwise, which is the same as 90 degrees counterclockwise.
C: This treats the 270-degree turn as if it were 180 degrees.
D: The counterclockwise turn cancels only the first 90 degrees, not the 270-degree turn.
Question 12 of 39 · medium
On a 4 by 4 grid (rows from the top, columns from the left), a shaded tile sits at (1, 2). The whole grid turns 90 degrees clockwise. Where is the tile now?
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Answer: B
Correct answer: (2, 4). Use the clockwise rule with n = 4:
(r,c)→(c,n+1−r)
The new row is the old column, 2, and the new column is
4 + 1 - 1 = 4
so the tile is at (2, 4).
A: This uses the counterclockwise rule (r, c) to (n + 1 - c, r).
B (correct): The clockwise rule (r, c) to (c, n + 1 - r) with n = 4 gives (2, 4 + 1 - 1) = (2, 4).
C: This only swaps the row and column numbers, which is a flip over the main diagonal, not a turn.
D: This is the result of a half-turn, (n + 1 - r, n + 1 - c).
Question 13 of 39 · medium
A marked tile is in row 2, column 5 of a 5 by 5 grid whose rows count down from the top. If the grid makes a quarter-turn counterclockwise about its center, which cell (r, c) holds the marked tile afterward?
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Answer: A
Correct answer: (1, 2). For a counterclockwise quarter-turn, (r, c) goes to (n + 1 - c, r). With n = 5, the new row is
6 - 5 = 1
and the new column is the old row, 2, so the tile lands at (1, 2), in the top row.
A (correct): The counterclockwise rule (r, c) to (n + 1 - c, r) with n = 5 gives (6 - 5, 2) = (1, 2).
B: This is where a half-turn sends the tile, not a quarter-turn.
C: This applies the clockwise rule, turning the grid the wrong way.
D: This just swaps the row and column, which is a diagonal flip rather than a turn.
Question 14 of 39 · medium
With the center of rotation at the origin, the point (2, -3) is turned a quarter-turn counterclockwise. Where does it land?
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Answer: C
Correct answer: (3, 2). Apply the counterclockwise quarter-turn rule:
(x,y)→(−y,x)
With x = 2 and
y = -3
the image is (3, 2). Both points are 13 units from the origin, as a rotation requires.
A: This is the clockwise result, (x, y) to (y, -x).
B: This only flips the point over the x-axis, which is a reflection, not a turn.
C (correct): The counterclockwise rule (x, y) to (-y, x) gives (-(-3), 2) = (3, 2).
D: This is a half-turn, which changes the sign of both coordinates.
Question 15 of 39 · medium
Three landmarks on a piece read dot, notch, hole when you go around it clockwise. After a move, which clockwise reading proves the piece was flipped over?
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Answer: D
Correct answer: Notch, dot, hole. Turns and shifts keep the clockwise cycle, though the reading may start at a different landmark. Notch, dot, hole is the same cycle as dot, hole, notch, which is the original order reversed, so the piece was mirrored.
A: This is the same cycle as dot, notch, hole, just starting at the notch, so a turn could produce it.
B: This is the original cycle starting at the hole, which a turn can produce.
C: This is the original order, which fits a turn or a shift, not a flip.
D (correct): This reverses the cycle (it equals dot, hole, notch), and only a reflection reverses clockwise order.
Question 16 of 39 · medium
The pieces are a 2 by 3 rectangle, a right triangle with legs of 4 and 3 units, and a 2 by 2 square. Which finished outline passes the area check?
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Answer: C
Correct answer: A 4 by 4 square. Add the piece areas: 2 times 3 is 6, half of 4 times 3 is 6, and 2 times 2 is 4, for a total of 16 square units. Only the 4 by 4 square has an area of 16.
A: Its area is 15 square units, one less than the 16 square units of the pieces.
B: Its area is 14 square units, which is too small for 16 square units of pieces.
C (correct): Its area is 16 square units, matching the total 6 + 6 + 4 = 16 of the pieces.
D: Its area is 18 square units, two more than the pieces can cover, so gaps would remain.
Question 17 of 39 · medium
The point (1, 3) is turned a quarter-turn counterclockwise about the origin and then moved 2 units down. Where does it end up?
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Answer: B
Correct answer: (-3, -1). Step 1: the counterclockwise rule (x, y) to (-y, x) sends (1, 3) to (-3, 1).
Step 2: moving 2 units down subtracts 2 from y, giving (-3, -1).
A: This does the moves in the wrong order: moving down first gives (1, 1), and turning that gives (-1, 1).
B (correct): Turning first gives (-3, 1), and moving 2 units down then gives (-3, -1).
C: This uses a clockwise turn, (1, 3) to (3, -1), before moving down.
D: This makes a half-turn instead of a quarter-turn: (1, 3) goes to (-1, -3), and moving 2 units down gives (-1, -5).
Question 18 of 39 · medium
You face south. You make three quarter-turns counterclockwise and then one half-turn. Which way do you face, and where does your left hand point?
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Answer: D
Correct answer: Facing east, left hand pointing north. Count counterclockwise as negative:
-270 + 180 = -90
which is one quarter-turn counterclockwise. Counterclockwise from south is east, and when you face east your left hand points north.
A: The facing is right, but south is on the right-hand side when facing east, not the left.
B: This keeps only the half-turn and treats the three quarter-turns as if they cancelled out.
C: This ignores the final half-turn; three counterclockwise quarter-turns from south alone end at west.
D (correct): The net turn is -270 + 180 = -90, a quarter-turn counterclockwise, which takes south to east, and facing east your left is north.
Question 19 of 39 · medium
A right triangle with legs of 5 and 12 units and a hypotenuse of 13 units must join a 13 by 4 rectangle along their only matching edge. What is the area of the finished figure?
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Answer: A
Correct answer: 82 square units. The only matching length is 13, so the hypotenuse meets the 13-unit side. Area: one-half of 5 times 12 is 30, plus 13 times 4 is 52, giving 82 square units.
A (correct): The triangle has area one-half of 5 times 12, which is 30, and the rectangle has 13 times 4 = 52, for a total of 82.
B: This is the area of the rectangle alone; the triangle's area is missing.
C: This halves the whole total, applying the one-half to the rectangle as well.
D: This forgets the one-half in the triangle area, using 5 times 12 = 60 instead of 30.
Question 20 of 39 · medium
A grid has 3 rows and 5 columns, numbered from the top and from the left. A tile at (1, 4) is turned with the grid 90 degrees clockwise, using the rule (r, c) to (c, m + 1 - r), where m is the number of rows. Where is the tile?
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Answer: D
Correct answer: (4, 3). For an m-row grid, a clockwise quarter-turn sends (r, c) to (c, m + 1 - r). Here m = 3, so (1, 4) goes to
(4, 3 + 1 - 1) = (4, 3)
and the turned grid has 5 rows and 3 columns.
A: This applies a counterclockwise rule, turning the grid the wrong way.
B: This uses the column count, 5, in place of the row count, 3, in the rule.
C: This is a left-right flip of the tile, not a turn.
D (correct): With m = 3, the new row is the old column, 4, and the new column is 3 + 1 - 1 = 3.
Question 21 of 39 · medium
The source picture shows a rigid piece on a 5-by-5 grid, with rows counted from the top and columns from the left. Its tiles are at (1, 1), (1, 2), (2, 2), (3, 2), (3, 3), (4, 3), and the orange dotted marker tile is at (1, 1). The piece is given a 90-degree clockwise turn about the center of the grid, and the marker moves with it. Which picture shows the result?
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Answer: B
Use the rule
(r,c)→(c,6−r)
The marker at (1, 1) moves to (1, 5), and the landmark tile at (4, 3) moves to (3, 2). Only one picture has the marker at (1, 5) together with a tile at (3, 2), and its remaining tiles also match the rule.
A: This shows a 90-degree counterclockwise turn: the marker sits at (5, 1), where that move sends (1, 1), instead of at (1, 5).
B (correct): This picture follows (r,c)→(c,6−r) for every tile: the marker lands at (1, 5) and the tile from (4, 3) lands at (3, 2), with the shape unchanged.
C: This shows a left-right mirror across the vertical centerline: the marker happens to land at (1, 5), but the tile from (4, 3) sits at (4, 3) instead of (3, 2). It is a mirror image, so the landmarks' clockwise order is reversed.
D: This shows a 180-degree half-turn: the marker sits at (5, 5), where that move sends (1, 1), instead of at (1, 5).
Question 22 of 39 · medium
Picture the source piece on its 5-by-5 grid: tiles (1, 4), (2, 3), (2, 4), (3, 3), (4, 2), (4, 3), with the orange dotted marker at (4, 2) (row first, rows numbered from the top). Apply exactly a 90-degree counterclockwise turn, keeping the piece rigid and carrying the marker tile along. Which picture shows where every tile lands?
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Answer: D
Use the rule
(r,c)→(6−c,r)
The marker at (4, 2) moves to (4, 4), and the landmark tile at (1, 4) moves to (2, 1). Only one picture has the marker at (4, 4) together with a tile at (2, 1), and its remaining tiles also match the rule.
A: This shows a 180-degree half-turn: the marker sits at (2, 4), where that move sends (4, 2), instead of at (4, 4).
B: This shows a 90-degree clockwise turn: the marker sits at (2, 2), where that move sends (4, 2), instead of at (4, 4).
C: This shows a left-right mirror across the vertical centerline: the marker happens to land at (4, 4), but the tile from (1, 4) sits at (1, 2) instead of (2, 1). It is a mirror image, so the landmarks' clockwise order is reversed.
D (correct): This picture follows (r,c)→(6−c,r) for every tile: the marker lands at (4, 4) and the tile from (1, 4) lands at (2, 1), with the shape unchanged.
Question 23 of 39 · medium
A rigid piece fills cells (1, 2), (2, 2), (3, 2), (3, 3), (3, 4), (4, 4) of a 5-by-5 grid (row, column, counted from the top-left), and its orange dotted marker tile sits at (1, 2). Which picture shows this piece after a 180-degree turn, with the marker carried along?
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Answer: A
Use the rule
(r,c)→(6−r,6−c)
The marker at (1, 2) moves to (5, 4), and the landmark tile at (4, 4) moves to (2, 2). Only one picture has the marker at (5, 4) together with a tile at (2, 2), and its remaining tiles also match the rule.
A (correct): This picture follows (r,c)→(6−r,6−c) for every tile: the marker lands at (5, 4) and the tile from (4, 4) lands at (2, 2), with the shape unchanged.
B: This shows a left-right mirror across the vertical centerline: the marker sits at (1, 4), where that move sends (1, 2), instead of at (5, 4). It is a mirror image, so the landmarks' clockwise order is reversed.
C: This shows a 90-degree clockwise turn: the marker sits at (2, 5), where that move sends (1, 2), instead of at (5, 4).
D: This shows a 90-degree counterclockwise turn: the marker sits at (4, 1), where that move sends (1, 2), instead of at (5, 4).
Question 24 of 39 · medium
Using row-column positions, the source piece covers (1, 1), (2, 1), (2, 2), (3, 2), (4, 2), (4, 3) and has its orange dotted marker at (2, 1). If the whole piece undergoes a left-right mirror flip across the vertical centerline and nothing else changes, which picture shows the outcome?
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Answer: C
Use the rule
(r,c)→(r,6−c)
The marker at (2, 1) moves to (2, 5), and the landmark tile at (1, 1) moves to (1, 5). Only one picture has the marker at (2, 5) together with a tile at (1, 5), and its remaining tiles also match the rule. Because the move is a mirror, the clockwise order of the landmarks reverses, so no plain turn can produce this picture.
A: This shows a 90-degree counterclockwise turn: the marker sits at (5, 2), where that move sends (2, 1), instead of at (2, 5). A turn keeps the clockwise order of landmarks, so it cannot give the required mirror image.
B: This shows a 90-degree clockwise turn: the marker sits at (1, 4), where that move sends (2, 1), instead of at (2, 5). A turn keeps the clockwise order of landmarks, so it cannot give the required mirror image.
C (correct): This picture follows (r,c)→(r,6−c) for every tile: the marker lands at (2, 5) and the tile from (1, 1) lands at (1, 5), with the shape unchanged.
D: This shows a 180-degree half-turn: the marker sits at (4, 5), where that move sends (2, 1), instead of at (2, 5). A turn keeps the clockwise order of landmarks, so it cannot give the required mirror image.
Question 25 of 39 · medium
Look at the source piece: 6 tiles at (1, 3), (2, 3), (3, 3), (3, 4), (4, 4), (4, 5), marker at (4, 5), on a 5-by-5 grid numbered from the top-left corner. The piece cannot bend, and the orange dotted marker stays attached to it. After a top-bottom mirror flip across the horizontal centerline, which picture is correct?
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Answer: A
Use the rule
(r,c)→(6−r,c)
The marker at (4, 5) moves to (2, 5), and the landmark tile at (1, 3) moves to (5, 3). Only one picture has the marker at (2, 5) together with a tile at (5, 3), and its remaining tiles also match the rule. Because the move is a mirror, the clockwise order of the landmarks reverses, so no plain turn can produce this picture.
A (correct): This picture follows (r,c)→(6−r,c) for every tile: the marker lands at (2, 5) and the tile from (1, 3) lands at (5, 3), with the shape unchanged.
B: This shows a 180-degree half-turn: the marker sits at (2, 1), where that move sends (4, 5), instead of at (2, 5). A turn keeps the clockwise order of landmarks, so it cannot give the required mirror image.
C: This shows a 90-degree clockwise turn: the marker sits at (5, 2), where that move sends (4, 5), instead of at (2, 5). A turn keeps the clockwise order of landmarks, so it cannot give the required mirror image.
D: This shows a 90-degree counterclockwise turn: the marker sits at (1, 4), where that move sends (4, 5), instead of at (2, 5). A turn keeps the clockwise order of landmarks, so it cannot give the required mirror image.
Question 26 of 39 · medium
Suppose you perform a 90-degree clockwise turn on the piece in the source picture. Before the move, its 6 cells are (1, 5), (2, 5), (3, 3), (3, 4), (3, 5), (4, 3) (rows from the top, columns from the left), with the orange dotted tile at (1, 5). Which answer picture matches the moved piece?
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Answer: A
Use the rule
(r,c)→(c,6−r)
The marker at (1, 5) moves to (5, 5), and the landmark tile at (3, 5) moves to (5, 3). Only one picture has the marker at (5, 5) together with a tile at (5, 3), and its remaining tiles also match the rule.
A (correct): This picture follows (r,c)→(c,6−r) for every tile: the marker lands at (5, 5) and the tile from (3, 5) lands at (5, 3), with the shape unchanged.
B: This shows a 180-degree half-turn: the marker sits at (5, 1), where that move sends (1, 5), instead of at (5, 5).
C: This shows a left-right mirror across the vertical centerline: the marker sits at (1, 1), where that move sends (1, 5), instead of at (5, 5). It is a mirror image, so the landmarks' clockwise order is reversed.
D: This shows a 90-degree counterclockwise turn: the marker sits at (1, 1), where that move sends (1, 5), instead of at (5, 5).
Question 27 of 39 · medium
Which picture is the source piece after a 90-degree counterclockwise turn? Treat the piece as one solid part; the orange dotted marker, which starts at (3, 3), travels with it. The starting cells, written (row, column) on the 5-by-5 grid, are (1, 1), (2, 1), (3, 1), (3, 2), (3, 3), (4, 3).
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Answer: C
Use the rule
(r,c)→(6−c,r)
The marker at (3, 3) is the center cell, so it stays at (3, 3) under every turn and mirror and cannot decide the answer by itself. Follow the landmark tile at (1, 1) instead: it must land at (5, 1). Only one picture has the marker at the center and a tile at (5, 1), and its other tiles also follow the rule.
A: This shows a 180-degree half-turn: the marker happens to land at (3, 3), but the tile from (1, 1) sits at (5, 5) instead of (5, 1).
B: This shows a 90-degree clockwise turn: the marker happens to land at (3, 3), but the tile from (1, 1) sits at (1, 5) instead of (5, 1).
C (correct): This picture follows (r,c)→(6−c,r) for every tile: the marker lands at (3, 3) and the tile from (1, 1) lands at (5, 1), with the shape unchanged.
D: This shows a left-right mirror across the vertical centerline: the marker happens to land at (3, 3), but the tile from (1, 1) sits at (1, 5) instead of (5, 1). It is a mirror image, so the landmarks' clockwise order is reversed.
Question 28 of 39 · medium
Task: carry out a left-right mirror flip across the vertical centerline. Grid: 5 by 5, row 1 at the top, column 1 at the left. Piece: (2, 1), (2, 2), (3, 2), (3, 3), (4, 3), (5, 3), orange dotted marker at (5, 3), rigid. Which picture shows the finished position?
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Answer: A
Use the rule
(r,c)→(r,6−c)
The marker at (5, 3) moves to (5, 3), and the landmark tile at (2, 1) moves to (2, 5). Only one picture has the marker at (5, 3) together with a tile at (2, 5), and its remaining tiles also match the rule. Because the move is a mirror, the clockwise order of the landmarks reverses, so no plain turn can produce this picture.
A (correct): This picture follows (r,c)→(r,6−c) for every tile: the marker lands at (5, 3) and the tile from (2, 1) lands at (2, 5), with the shape unchanged.
B: This shows a 90-degree clockwise turn: the marker sits at (3, 1), where that move sends (5, 3), instead of at (5, 3). A turn keeps the clockwise order of landmarks, so it cannot give the required mirror image.
C: This shows a 180-degree half-turn: the marker sits at (1, 3), where that move sends (5, 3), instead of at (5, 3). A turn keeps the clockwise order of landmarks, so it cannot give the required mirror image.
D: This shows a 90-degree counterclockwise turn: the marker sits at (3, 5), where that move sends (5, 3), instead of at (5, 3). A turn keeps the clockwise order of landmarks, so it cannot give the required mirror image.
Question 29 of 39 · medium
The orange dotted marker of the source piece is at (1, 3), and the rest of the piece occupies the other cells of (1, 2), (1, 3), (2, 3), (3, 3), (3, 4), (4, 4) on a 5-by-5 grid. Every tile, marker included, is moved by a 180-degree turn at once. Which picture results?
Show the answer
Answer: B
Use the rule
(r,c)→(6−r,6−c)
The marker at (1, 3) moves to (5, 3), and the landmark tile at (4, 4) moves to (2, 2). Only one picture has the marker at (5, 3) together with a tile at (2, 2), and its remaining tiles also match the rule.
A: This shows a left-right mirror across the vertical centerline: the marker sits at (1, 3), where that move sends (1, 3), instead of at (5, 3). It is a mirror image, so the landmarks' clockwise order is reversed.
B (correct): This picture follows (r,c)→(6−r,6−c) for every tile: the marker lands at (5, 3) and the tile from (4, 4) lands at (2, 2), with the shape unchanged.
C: This shows a 90-degree counterclockwise turn: the marker sits at (3, 1), where that move sends (1, 3), instead of at (5, 3).
D: This shows a 90-degree clockwise turn: the marker sits at (3, 5), where that move sends (1, 3), instead of at (5, 3).
Question 30 of 39 · hard
On the 5-by-5 grid, the source piece is made of 7 tiles: (1, 3), (2, 3), (3, 3), (4, 1), (4, 2), (4, 3), (5, 1). One of them, (4, 1), carries the orange dotted marker. Choose the picture that shows exactly a 90-degree clockwise turn of this piece. Which one is it?
Show the answer
Answer: A
Use the rule
(r,c)→(c,6−r)
The marker at (4, 1) moves to (1, 2), and the landmark tile at (1, 3) moves to (3, 5). Only one picture has the marker at (1, 2) together with a tile at (3, 5), and its remaining tiles also match the rule.
A (correct): This picture follows (r,c)→(c,6−r) for every tile: the marker lands at (1, 2) and the tile from (1, 3) lands at (3, 5), with the shape unchanged.
B: This shows a 180-degree half-turn: the marker sits at (2, 5), where that move sends (4, 1), instead of at (1, 2).
C: This shows a left-right mirror across the vertical centerline: the marker sits at (4, 5), where that move sends (4, 1), instead of at (1, 2). It is a mirror image, so the landmarks' clockwise order is reversed.
D: This shows a 90-degree counterclockwise turn: the marker sits at (5, 4), where that move sends (4, 1), instead of at (1, 2).
Question 31 of 39 · hard
Imagine the 5-by-5 grid as a card that you can move. The source piece sits on cells (1, 1), (1, 2), (2, 2), (3, 2), (3, 3), (4, 3), (5, 3), its orange dotted marker on (1, 1). Give the card a 180-degree turn. Which picture shows the piece now?
Show the answer
Answer: C
Use the rule
(r,c)→(6−r,6−c)
The marker at (1, 1) moves to (5, 5), and the landmark tile at (5, 3) moves to (1, 3). Only one picture has the marker at (5, 5) together with a tile at (1, 3), and its remaining tiles also match the rule.
A: This shows a 90-degree clockwise turn: the marker sits at (1, 5), where that move sends (1, 1), instead of at (5, 5).
B: This shows a 90-degree counterclockwise turn: the marker sits at (5, 1), where that move sends (1, 1), instead of at (5, 5).
C (correct): This picture follows (r,c)→(6−r,6−c) for every tile: the marker lands at (5, 5) and the tile from (5, 3) lands at (1, 3), with the shape unchanged.
D: This shows a left-right mirror across the vertical centerline: the marker sits at (1, 5), where that move sends (1, 1), instead of at (5, 5). It is a mirror image, so the landmarks' clockwise order is reversed.
Question 32 of 39 · hard
What does the source piece look like after a top-bottom mirror flip across the horizontal centerline? It covers (1, 5), (2, 4), (2, 5), (3, 4), (4, 3), (4, 4), (5, 3) on a 5-by-5 grid, where (r, c) means row r from the top and column c from the left, and its orange dotted marker tile is (2, 4).
Show the answer
Answer: C
Use the rule
(r,c)→(6−r,c)
The marker at (2, 4) moves to (4, 4), and the landmark tile at (2, 5) moves to (4, 5). Only one picture has the marker at (4, 4) together with a tile at (4, 5), and its remaining tiles also match the rule. Because the move is a mirror, the clockwise order of the landmarks reverses, so no plain turn can produce this picture.
A: This shows a 180-degree half-turn: the marker sits at (4, 2), where that move sends (2, 4), instead of at (4, 4). A turn keeps the clockwise order of landmarks, so it cannot give the required mirror image.
B: This shows a 90-degree clockwise turn: the marker happens to land at (4, 4), but the tile from (2, 5) sits at (5, 4) instead of (4, 5). A turn keeps the clockwise order of landmarks, so it cannot give the required mirror image.
C (correct): This picture follows (r,c)→(6−r,c) for every tile: the marker lands at (4, 4) and the tile from (2, 5) lands at (4, 5), with the shape unchanged.
D: This shows a 90-degree counterclockwise turn: the marker sits at (2, 2), where that move sends (2, 4), instead of at (4, 4). A turn keeps the clockwise order of landmarks, so it cannot give the required mirror image.
Question 33 of 39 · hard
Starting cells of the rigid source piece: (1, 2), (2, 2), (3, 2), (3, 3), (4, 3), (5, 3), (5, 4). Marker (orange, dotted): (1, 2). The piece receives a left-right mirror flip across the vertical centerline on the 5-by-5 grid and keeps its shape. Pick the picture showing the new position. Which is it?
Show the answer
Answer: A
Use the rule
(r,c)→(r,6−c)
The marker at (1, 2) moves to (1, 4), and the landmark tile at (5, 4) moves to (5, 2). Only one picture has the marker at (1, 4) together with a tile at (5, 2), and its remaining tiles also match the rule. Because the move is a mirror, the clockwise order of the landmarks reverses, so no plain turn can produce this picture.
A (correct): This picture follows (r,c)→(r,6−c) for every tile: the marker lands at (1, 4) and the tile from (5, 4) lands at (5, 2), with the shape unchanged.
B: This shows a 90-degree counterclockwise turn: the marker sits at (4, 1), where that move sends (1, 2), instead of at (1, 4). A turn keeps the clockwise order of landmarks, so it cannot give the required mirror image.
C: This shows a 90-degree clockwise turn: the marker sits at (2, 5), where that move sends (1, 2), instead of at (1, 4). A turn keeps the clockwise order of landmarks, so it cannot give the required mirror image.
D: This shows a 180-degree half-turn: the marker sits at (5, 4), where that move sends (1, 2), instead of at (1, 4). A turn keeps the clockwise order of landmarks, so it cannot give the required mirror image.
Question 34 of 39 · hard
Each tile of the source piece, including the orange dotted marker at (5, 2), is moved by the same rule: a 90-degree counterclockwise turn of the 5-by-5 grid. The tiles begin at (1, 4), (2, 4), (3, 3), (3, 4), (4, 3), (5, 2), (5, 3). Which picture shows them after the move?
Show the answer
Answer: C
Use the rule
(r,c)→(6−c,r)
The marker at (5, 2) moves to (4, 5), and the landmark tile at (4, 3) moves to (3, 4). Only one picture has the marker at (4, 5) together with a tile at (3, 4), and its remaining tiles also match the rule.
A: This shows a left-right mirror across the vertical centerline: the marker sits at (5, 4), where that move sends (5, 2), instead of at (4, 5). It is a mirror image, so the landmarks' clockwise order is reversed.
B: This shows a 90-degree clockwise turn: the marker sits at (2, 1), where that move sends (5, 2), instead of at (4, 5).
C (correct): This picture follows (r,c)→(6−c,r) for every tile: the marker lands at (4, 5) and the tile from (4, 3) lands at (3, 4), with the shape unchanged.
D: This shows a 180-degree half-turn: the marker sits at (1, 4), where that move sends (5, 2), instead of at (4, 5).
Question 35 of 39 · hard
Here the only allowed change is a 90-degree clockwise turn. The source piece occupies (1, 2), (2, 2), (3, 1), (3, 2), (4, 2), (5, 2), (5, 3) on a 5-by-5 grid with row 1 on top, and the orange dotted marker is on (3, 1). Which picture shows the piece after that change and nothing else?
Show the answer
Answer: B
Use the rule
(r,c)→(c,6−r)
The marker at (3, 1) moves to (1, 3), and the landmark tile at (5, 3) moves to (3, 1). Only one picture has the marker at (1, 3) together with a tile at (3, 1), and its remaining tiles also match the rule.
A: This shows a 90-degree counterclockwise turn: the marker sits at (5, 3), where that move sends (3, 1), instead of at (1, 3).
B (correct): This picture follows (r,c)→(c,6−r) for every tile: the marker lands at (1, 3) and the tile from (5, 3) lands at (3, 1), with the shape unchanged.
C: This shows a left-right mirror across the vertical centerline: the marker sits at (3, 5), where that move sends (3, 1), instead of at (1, 3). It is a mirror image, so the landmarks' clockwise order is reversed.
D: This shows a 180-degree half-turn: the marker sits at (3, 5), where that move sends (3, 1), instead of at (1, 3).
Question 36 of 39 · hard
A grid card holds the source piece in cells (1, 5), (2, 5), (3, 4), (3, 5), (4, 3), (4, 4), (5, 3), with the orange dotted marker tile at (3, 4). Someone applies a left-right mirror flip across the vertical centerline to the card. Which picture shows what they see afterward?
Show the answer
Answer: A
Use the rule
(r,c)→(r,6−c)
The marker at (3, 4) moves to (3, 2), and the landmark tile at (2, 5) moves to (2, 1). Only one picture has the marker at (3, 2) together with a tile at (2, 1), and its remaining tiles also match the rule. Because the move is a mirror, the clockwise order of the landmarks reverses, so no plain turn can produce this picture.
A (correct): This picture follows (r,c)→(r,6−c) for every tile: the marker lands at (3, 2) and the tile from (2, 5) lands at (2, 1), with the shape unchanged.
B: This shows a 180-degree half-turn: the marker happens to land at (3, 2), but the tile from (2, 5) sits at (4, 1) instead of (2, 1). A turn keeps the clockwise order of landmarks, so it cannot give the required mirror image.
C: This shows a 90-degree counterclockwise turn: the marker sits at (2, 3), where that move sends (3, 4), instead of at (3, 2). A turn keeps the clockwise order of landmarks, so it cannot give the required mirror image.
D: This shows a 90-degree clockwise turn: the marker sits at (4, 3), where that move sends (3, 4), instead of at (3, 2). A turn keeps the clockwise order of landmarks, so it cannot give the required mirror image.
Question 37 of 39 · hard
Follow the source piece through a 90-degree counterclockwise turn. On its 5-by-5 grid (rows from the top) the piece covers (1, 1), (2, 1), (2, 2), (3, 2), (4, 2), (4, 3), (5, 3); the dotted orange marker is the tile at (2, 1). Which picture is the piece's image?
Show the answer
Answer: A
Use the rule
(r,c)→(6−c,r)
The marker at (2, 1) moves to (5, 2), and the landmark tile at (5, 3) moves to (3, 5). Only one picture has the marker at (5, 2) together with a tile at (3, 5), and its remaining tiles also match the rule.
A (correct): This picture follows (r,c)→(6−c,r) for every tile: the marker lands at (5, 2) and the tile from (5, 3) lands at (3, 5), with the shape unchanged.
B: This shows a 180-degree half-turn: the marker sits at (4, 5), where that move sends (2, 1), instead of at (5, 2).
C: This shows a 90-degree clockwise turn: the marker sits at (1, 4), where that move sends (2, 1), instead of at (5, 2).
D: This shows a left-right mirror across the vertical centerline: the marker sits at (2, 5), where that move sends (2, 1), instead of at (5, 2). It is a mirror image, so the landmarks' clockwise order is reversed.
Question 38 of 39 · hard
Of the four pictures, which shows the source piece moved by a 180-degree turn on the 5-by-5 grid? The original tiles are (1, 4), (2, 3), (2, 4), (3, 3), (4, 2), (4, 3), (5, 2), and the orange dotted marker starts at (1, 4), staying fixed to the piece.
Show the answer
Answer: B
Use the rule
(r,c)→(6−r,6−c)
The marker at (1, 4) moves to (5, 2), and the landmark tile at (5, 2) moves to (1, 4). Only one picture has the marker at (5, 2) together with a tile at (1, 4), and its remaining tiles also match the rule. The outline after this move covers the same cells as the original, so the marker is what tells the correct picture apart from an unmoved copy.
A: This shows a 90-degree clockwise turn: the marker sits at (4, 5), where that move sends (1, 4), instead of at (5, 2).
B (correct): This picture follows (r,c)→(6−r,6−c) for every tile: the marker lands at (5, 2) and the tile from (5, 2) lands at (1, 4), with the shape unchanged.
C: This shows a 90-degree counterclockwise turn: the marker sits at (2, 1), where that move sends (1, 4), instead of at (5, 2).
D: This shows a left-right mirror across the vertical centerline: the marker sits at (1, 2), where that move sends (1, 4), instead of at (5, 2). It is a mirror image, so the landmarks' clockwise order is reversed.
Question 39 of 39 · hard
The source piece's cells are (1, 2), (2, 2), (3, 2), (3, 3), (4, 3), (4, 4), (5, 4) in (row, column) form, rows counted downward. Its orange dotted marker is at (4, 4). Under a top-bottom mirror flip across the horizontal centerline, which picture shows where the piece ends up?
Show the answer
Answer: D
Use the rule
(r,c)→(6−r,c)
The marker at (4, 4) moves to (2, 4), and the landmark tile at (1, 2) moves to (5, 2). Only one picture has the marker at (2, 4) together with a tile at (5, 2), and its remaining tiles also match the rule. Because the move is a mirror, the clockwise order of the landmarks reverses, so no plain turn can produce this picture.
A: This shows a 180-degree half-turn: the marker sits at (2, 2), where that move sends (4, 4), instead of at (2, 4). A turn keeps the clockwise order of landmarks, so it cannot give the required mirror image.
B: This shows a 90-degree counterclockwise turn: the marker happens to land at (2, 4), but the tile from (1, 2) sits at (4, 1) instead of (5, 2). A turn keeps the clockwise order of landmarks, so it cannot give the required mirror image.
C: This shows a 90-degree clockwise turn: the marker sits at (4, 2), where that move sends (4, 4), instead of at (2, 4). A turn keeps the clockwise order of landmarks, so it cannot give the required mirror image.
D (correct): This picture follows (r,c)→(6−r,c) for every tile: the marker lands at (2, 4) and the tile from (1, 2) lands at (5, 2), with the shape unchanged.
Each answer is followed by an explanation that describes the move step by step, which is useful because spatial errors are hard to diagnose on your own. If mirrored answers keep fooling you, return to the handedness and reflection-disguised-as-rotation sections. If you lose track during two-step moves, revisit the section on reducing complex motion into small turns and the two-round checking strategy.
Your score on this page is a practice measure only, not an ASVAB or AFQT result. Once you are comfortable here, continue to the Matching Connection Points lesson.
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