Which of the pieces shown completes the pattern? (The five choices A–E are pictured below the question.)
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Answer: C
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Hint 1 of 2
The big design is one repeating pattern; the white window is just a square-shaped hole punched out of it.
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Hint 2 of 2
Look at the lines touching all four edges of the hole and ask which piece lets every one of them continue without a break.
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Approach: match the missing tile to the lines around the hole
The hole sits inside a repeating pattern of overlapping squares and diamonds, so the right piece is the one that keeps every line going straight across the gap.
Trace the lines that arrive at the top, bottom, left and right edges of the white square; the correct piece must connect to all of them at once.
Only choice C lines up on all four sides so the pattern stays seamless with no broken lines, so the answer is (C).
Anna builds a wall out of black and grey bricks that shows 2025. What can Bella read on the back of the wall? (The five choices A–E are pictured below the question.)
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Answer: E
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Hint 1 of 2
Looking at the back of a wall is like seeing it in a mirror.
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Hint 2 of 2
Reflect the whole ‘2025’ left-to-right; each digit flips and the order of digits reverses.
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Approach: horizontal mirror reflection
Seeing the back of the wall is exactly like holding the front up to a mirror, so the whole picture flips left–right.
Two things happen at once: the order of the digits reverses (so 2025 reads 5202), and each digit itself is mirrored.
The choice that shows this left–right flip of the bricks is the back view, which is (E).
Thea rotates a painted hexagon clockwise one space at a time. The first rotation can be seen in the picture. Which hexagon does Thea see after the eighth rotation? (The five choices A–E are pictured below the question.)
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Answer: A
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Hint 1 of 2
The hexagon has 6 sectors, so rotating 6 times brings it back to the start.
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Hint 2 of 2
Eight rotations is the same as just 8 − 6 = 2 rotations.
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Approach: rotation repeats every 6 steps
A hexagon has 6 sectors, so after 6 one-step turns it looks exactly like the start — the pattern repeats every 6 rotations.
So the 8th rotation looks the same as the 8 − 6 = 2nd rotation; we only need to turn the starting hexagon two steps clockwise.
Turning the Start picture two sectors clockwise matches choice (A), so that is what Thea sees after the eighth rotation.
Holger writes the numbers up to 40 into the table in the same way as shown. Which of the pieces A to E can he then cut out from the table?
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Answer: C
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Hint 1 of 2
The numbers fill the table eight to a row: 1-8, then 9-16, then 17-24, and so on.
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Hint 2 of 2
Pin down where 12 sits, then check that the two cells just below it really hold 20 and 21, and that the cell hanging beneath them matches the piece's shape.
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Approach: locate the cells by their row-of-eight positions and match the exact shape
Each row holds eight numbers, so 12 sits in the second row, fourth column.
Directly under 12 are 20 then 21 (third row), and beneath 21 sits 29 (fourth row).
The piece whose three cells read 12 on top, 20 and 21 in the middle, and 29 hanging under the 21 is the only one whose outline matches these real positions.
Nine steps of a staircase winding around a cylinder can be seen, starting at the bottom and leading all the way to the top. All the steps are equally high. How many steps cannot be seen?
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Answer: D — 12
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Hint 1 of 3
From the front you only see the steps facing you; the rest of the staircase keeps winding around the hidden back of the cylinder.
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Hint 2 of 3
The picture shows the front-facing steps spiralling up; figure out how high the whole tower climbs, then take away the nine you can already see.
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Hint 3 of 3
Each level of the spiral has steps on both the front and the back, so the hidden back steps roughly mirror the visible front ones, plus a few more for the extra height.
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Approach: see that the spiral has front and back steps at each level, then count the hidden ones
The nine steps you can see are the ones facing you as the staircase spirals up the front of the cylinder.
As the spiral turns, the same number of steps run around the hidden back at each level, and the tower keeps climbing past where the front steps stop.
Counting the back steps level by level all the way to the top gives twelve steps that are turned away and cannot be seen.
Six points are placed and numbered as shown on the right. Two triangles are drawn: one by connecting the even-numbered points, and one by connecting the odd-numbered points. Which of the following shapes is the result?
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Answer: E
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Hint 1 of 2
Mark which of the six points are odd (1,3,5) and which are even (2,4,6), then picture the two triangles they make.
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Hint 2 of 2
Each triangle is fixed by its three points; overlay them and compare the combined outline to each option.
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Approach: connect the odd and even points and match the overlaid figure
The odd points 1,3,5 form one triangle and the even points 2,4,6 form another.
Drawing both on the given point positions, the two triangles cross each other in a particular way.
Comparing that overlap with the choices, only E reproduces it.
The two-sided mirrors reflect the laser beam as shown in the small picture on the left. At which letter does the laser beam leave the picture on the right?
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Answer: B — B
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Hint 1 of 2
Each diagonal mirror turns the beam by a right angle; the small example shows which way.
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Hint 2 of 2
Step the beam square by square, bouncing 90 degrees at every mirror, until it reaches an edge.
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Approach: trace the beam, reflecting 90 degrees at each mirror
Use the small picture to learn how each slanted mirror deflects the beam.
Starting from the entry arrow on the big grid, advance the beam and turn it a quarter-turn at every mirror it hits.
Following the bounces, the beam leaves the grid at letter B.
Which of the following solid shapes can be made with these 6 bricks?
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Answer: D
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Hint 1 of 2
The 6 bricks are 2 white and 4 grey, and each brick is a 1x1x2 block.
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Hint 2 of 2
Build the answer from twelve unit cubes (4 white, 8 grey) and check which picture shows exactly that count of each colour with the brick seams in the right places.
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Approach: match brick colours and seams to a solid
Two white bricks and four grey bricks supply 4 white unit cubes and 8 grey unit cubes.
The assembled solid must show that 2:1 grey-to-white split, with the visible faces and the seams between bricks lining up.
Only choice D shows a solid whose colouring and brick seams can come from these six bricks.
In the square you can see the digits from 1 to 9. A number is created by starting at the star, following the line and writing down the digits along the line while passing. For example, the line shown represents the number 42685. Which of the following lines represents the largest number?
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Answer: E
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Hint 1 of 2
Read off the digit string each path makes, then compare them as numbers.
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Hint 2 of 2
The biggest number starts with the largest leading digit; break ties by the next digit.
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Approach: trace each path into a number and compare
Each option traces a path from the star across cells of the 1-9 grid, writing the digit of every cell it passes.
Convert each path to its number and compare digit by digit from the left.
Mary had a piece of paper. She folded it exactly in half. Then she folded it exactly in half again. She got the small shape shown on the left (a right triangle). Which of the shapes P, Q or R could have been the shape of her original piece of paper?
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Answer: E — any of P, Q or R
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Hint 1 of 2
Folding once then again maps the original onto a quarter-size shape; run it backwards.
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Hint 2 of 2
Unfold the right triangle twice and check which of P, Q, R it could grow back into.
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Approach: unfold the result twice
Folding a sheet in half twice can turn a rectangle, a square, or a larger right triangle into this small right triangle.
Unfolding the given triangle can recreate any of shapes P, Q or R, so the answer is E (any of P, Q or R).
Anna has four identical building blocks that each look like the one shown (a straight strip of three squares). Which of the shapes in the options can she not form with them?
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Answer: E
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Hint 1 of 2
Each block covers three squares in a straight line; four of them cover twelve squares.
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Hint 2 of 2
A shape can be built only if it can be cut into straight 1×3 pieces — try tiling each one.
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Approach: tile each shape with straight triominoes
The block is a straight strip of three squares, so four blocks cover 12 squares total.
Each pictured shape has 12 squares, so the test is whether it splits into four straight 1×3 strips.
Four of the shapes can be cut into such strips; the remaining one cannot be tiled this way.
Bob folds a piece of paper, then punches a hole in it and unfolds it again. The unfolded paper then looks like the picture. Along which dotted line did Bob fold the paper?
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Answer: D
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Hint 1 of 2
Unfolding mirrors the punched holes across each fold line, so the holes are symmetric about that line.
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Hint 2 of 2
Find the line that the four-hole pattern is symmetric across — that was the fold.
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Approach: match the hole pattern's symmetry to the fold line
A punch through folded paper leaves holes that are mirror images across the fold crease.
Look at the four holes in the unfolded sheet and find the line they are symmetric about.
The holes balance across the diagonal shown in (D).
A 10 cm long piece of wire is folded so that every part is equally long (see diagram). The wire is then cut through in the two marked positions. How long are the three pieces created in this way?
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Answer: A — 2 cm, 3 cm, 5 cm
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Hint 1 of 3
The wire is folded into equal little segments, so each segment is the same length.
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Hint 2 of 3
Imagine unfolding the wire into one straight 10 cm line and mark where the two cuts land.
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Hint 3 of 3
Count how many equal segments fall in each of the three pieces.
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Approach: unfold the wire and read off the cut positions
Folding 10 cm into equal parts makes a row of equal-length segments.
If you straighten the wire back out, the two marked cuts land on segment boundaries.
Counting the segments in each piece gives lengths of 2 cm, 3 cm and 5 cm (which add back to 10 cm), choice (A).
Four girls are sleeping in a room with their heads on the grey pillows. Bea and Pia are sleeping on the left-hand side of the room with their faces towards each other; Mary and Karen are on the right-hand side with their backs towards each other. How many girls sleep with their right ear on the pillow?
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Answer: C — 2
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Hint 1 of 3
For each girl, picture which cheek is pressed into the pillow and so which ear is underneath.
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Hint 2 of 3
When two girls lie side by side facing opposite ways, they rest on opposite ears.
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Hint 3 of 3
So in a face-to-face pair (and in a back-to-back pair) exactly one girl is on her right ear.
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Approach: pair up the girls and use mirror directions
Bea and Pia lie facing each other: since they point opposite ways, one rests on her left ear and the other on her right ear, so that pair gives 1 right-ear girl.
Mary and Karen lie back to back, again pointing opposite ways, so that pair also gives exactly 1 right-ear girl.
Adding the two pairs, \(1 + 1 = 2\) girls sleep on their right ear, choice (C).
For each shape, ask whether the grey part and the white part are the same size.
Still stuck? Show hint 2 →
Hint 2 of 2
An altitude of a triangle splits it into two pieces of equal area; check which figure is cut into two matching halves.
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Approach: compare grey area to whole in each picture
In the triangle the line goes straight down from the top vertex, cutting it into two pieces of equal area, and exactly one of them is grey.
The circle is in thirds (grey = one third), the four-square figure has three of four shaded, and the square-with-X and the pentagon-star are not split into two equal grey/white halves.
Only the triangle has exactly one half coloured grey, so the answer is B.
On the real umbrella the eight letters of KANGAROO appear in one fixed cyclic order around the rim.
Still stuck? Show hint 2 →
Hint 2 of 2
Pick the front letter of each pictured umbrella and read the neighbours left and right; the order around the rim must always match KANGAROO (just rotated).
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Approach: check the cyclic order of letters around each umbrella
Around the rim the letters always follow the same circular sequence K-A-N-G-A-R-O-O (the umbrella can only be turned, not rearranged).
Four of the pictures show that exact cyclic order, just rotated to a different front panel.
Picture C has the letters in an order that cannot be obtained by turning the umbrella, so it is the one that does not show the umbrella.
Arno lays out the word KANGAROO with 8 letter cards, but some cards are turned the wrong way (see picture). The letter K can be set right by turning its card twice, and the letter A by turning its card once. How many turns in all does Arno need so that KANGAROO reads correctly?
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Answer: C — 6
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Hint 1 of 2
Go letter by letter and decide whether each card is already the right way up.
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Hint 2 of 2
Count the cost: a letter that is upside-down or mirrored needs one or two turns to fix; add those up across the whole word.
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Approach: check each card and add up the turns it needs
Read the laid-out word against KANGAROO and find every card that is rotated or flipped.
Each wrong card needs either one turn or two turns to come right, exactly as the example shows for K and A.
Adding the turns needed across all the wrong cards gives a total of 6.
Nathalie wants to build a large cube out of small cubes (the complete cube is shown on the left). How many small cubes are missing from the shape on the right so that it would form the large cube?
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Answer: C — 7
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Hint 1 of 3
A full \(3\times3\times3\) cube is built from 27 little cubes.
Still stuck? Show hint 2 →
Hint 2 of 3
Count the little cubes already in the picture on the right, then see how many are still needed.
Still stuck? Show hint 3 →
Hint 3 of 3
Missing cubes = 27 minus the ones you counted.
Show solution
Approach: count present cubes and subtract from a full cube
The finished big cube is 3 cubes wide, 3 tall and 3 deep, so it needs \(3\times3\times3 = 27\) small cubes.
Counting the cubes in the picture on the right gives 20.
So the number still missing is \(27 - 20 = 7\), which is choice C.
In one move you may take some (or all) of the building blocks from the top of a stack, turn that group upside down, and put it back in the same place (see picture). Goran starts with the stack on the left and wants to end up with all the blocks ordered by size, as shown on the right. What is the smallest number of moves Goran needs?
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Answer: B — 3
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Hint 1 of 3
A move can only lift a group off the top and turn that whole group over, so the bottom blocks stay put unless you lift everything above them.
Still stuck? Show hint 2 →
Hint 2 of 3
Look at which blocks are already in the right size order and which clumps are reversed or out of place.
Still stuck? Show hint 3 →
Hint 3 of 3
Try to undo the disorder one reversed clump at a time, counting how few flips can finish the job.
Show solution
Approach: find the smallest set of top-flips that reorders the blocks
A move lifts some blocks off the top, turns that chunk over, and puts it back, so one move can reverse a top group.
Comparing the starting stack with the target size order shows which groups are out of place.
Carrying out the reorder with the fewest such flips takes 3 moves.
Tina draws shapes into each field of the pyramid. Each field in the second and third rows contains exactly the shapes of the two fields directly below it. Some fields are already filled in. Which shapes does she draw into the empty field of the bottom row?
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Answer: D
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Hint 1 of 3
A filled field is just the two fields below it combined, so a field above tells you the total of the pair underneath.
Still stuck? Show hint 2 →
Hint 2 of 3
Find a field whose value you know that sits right above the empty one, then subtract the shapes you can already see.
Still stuck? Show hint 3 →
Hint 3 of 3
Whatever shapes are missing after that subtraction must belong in the empty bottom field.
Show solution
Approach: use the rule that each field is the combination of the two below it
Each field equals the shapes of the two fields beneath it, so a middle field is the sum of its two bottom fields, and the top field is the sum of all three bottom fields (with the middle one counted twice).
Filling in the fields that are already given, subtract the known bottom fields from the totals to isolate the missing bottom field.
The shapes left over for the empty bottom field are one circle and one triangle.
Anna has two machines R and S. Machine R rotates a square piece of paper 90° clockwise (watch the marking in the corner). Machine S prints a club onto the paper. Anna wants to produce the picture shown. In which order does she use the two machines?
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Answer: B — RSRR
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Hint 1 of 3
Machine R only spins the paper a quarter-turn, while machine S stamps the club at whatever angle the paper is in right now.
Still stuck? Show hint 2 →
Hint 2 of 3
Keep your eye on the little corner marking and follow where it travels after each R turn.
Still stuck? Show hint 3 →
Hint 3 of 3
Read the orders one letter at a time, checking that the club gets stamped at the right moment so it ends up tilted the way the target shows.
Show solution
Approach: track the corner mark and the club's orientation through the machines
Follow the position of the corner marking as R turns the square 90° clockwise each time and S prints the club at the current orientation.
The target shows both the corner mark and the club in particular positions, so the printing must happen at the right stage and the later turns must carry both into place.
Testing the orders, R then S then R then R lands the marking and the club exactly as the target requires.
Monika wants to find a path through the maze from “Start” to “Ziel”. She may only move horizontally or vertically. She must enter every white circle exactly once and may not enter any black circle. In which direction must Monika move when she reaches the circle marked with x?
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Answer: A — ↓
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Hint 1 of 3
A circle in a corner or with black circles around it usually has only one open neighbour, so its move is forced.
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Hint 2 of 3
Start filling in those forced moves first, because each one locks in the next.
Still stuck? Show hint 3 →
Hint 3 of 3
By the time the forced path reaches x, only one direction keeps every remaining white circle reachable exactly once.
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Approach: use the must-visit-each-white-circle-once rule to force the path at x
Monika moves only horizontally and vertically, must enter every white circle exactly once, and cannot enter a black circle.
Near corners and beside black circles, several moves are forced because there is only one legal way through.
Tracing these forced moves up to the circle marked x leaves exactly one direction that still lets the path reach every remaining white circle: downward.
Some artwork is drawn on a square piece of transparent foil. The foil is folded over twice, as shown in the diagram. What does the foil look like after it has been folded over twice?
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Answer: A
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Hint 1 of 2
Each fold reflects the visible marks across the fold line onto the layer below.
Still stuck? Show hint 2 →
Hint 2 of 2
Fold once, draw where the marks land, then fold again and combine all layers.
Show solution
Approach: reflect the marks across each fold line
Folding flips the drawn marks across the fold crease onto the part beneath.
Apply the first fold, record the mirrored marks, then apply the second fold and overlay everything.
Anna has glued together several cubes of the same size to form a solid (see picture). Which of the following pictures shows a different view of this same solid?
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Answer: C
Show hints
Hint 1 of 2
Count the cubes and note the solid's overall shape, then mentally rotate it.
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Hint 2 of 2
A valid view must keep the same number of cubes and the same connections, just seen from another side.
Show solution
Approach: rotate the solid and match cube count and connections
The given solid has a fixed number of cubes joined in a particular way.
Each option is checked to see whether it is the same solid seen from a different direction.
Only C is a genuine rotation of the original solid.
The picture shows 3 gears with a black gear tooth on each. Which picture shows the correct position of the black teeth after the small gear has turned a full turn clockwise?
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Answer: A
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Hint 1 of 2
Meshed gears turn in opposite directions; a full turn of the small gear moves the others by matching tooth counts.
Still stuck? Show hint 2 →
Hint 2 of 2
Track the black tooth on each gear after that rotation to find the consistent picture.
Show solution
Approach: rotate each meshed gear correctly
When the small gear makes one full clockwise turn, the gears it meshes with rotate the other way by the same number of teeth.
Following each black tooth to its new position, the arrangement that results is choice A.
A large cube has side-length 7 cm. On each of its 6 faces, the two diagonals are drawn in red. The large cube is then cut into small cubes with side-length 1 cm. How many small cubes will have at least one red line drawn on it?
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Answer: B — 62
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Hint 1 of 2
A red face-diagonal only marks the unit cubes it passes through on that face; count by face then remove double counts.
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Hint 2 of 2
Edge and corner cubes can be crossed by diagonals on more than one face — don't count them twice.
Show solution
Approach: count marked cubes per face, then correct overlaps
Each face is a 7×7 grid of little squares; the two diagonals run through 7 + 7 − 1 = 13 of them (the centre square is shared).
Six faces give 6 × 13 = 78, but cubes along the edges and corners get a red line on two faces and were counted twice — there are 16 such double-counts.
So the number of unit cubes with at least one red line is 78 − 16 = 62.
There are rectangular cards divided into 4 equal cells with different shapes drawn in each cell. Cards can be placed side by side only if the same shapes appear in adjacent cells on their common side. 9 cards are used to form a rectangle as shown in the figure. Which of the following cards was definitely NOT used to form this rectangle?
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Answer: E
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Hint 1 of 2
Cards join only when the touching cells match, so trace the shape sequence along each row and column of the assembled rectangle.
Still stuck? Show hint 2 →
Hint 2 of 2
Read the forced shapes from the given grid; one listed card has a cell pattern that can never fit.
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Approach: match each card against the forced grid pattern
The assembled rectangle fixes which shapes sit in each cell because adjacent cards must agree on their shared edge.
Reading those forced shapes, four of the candidate cards can occur somewhere in the layout.
Card E has a cell arrangement that cannot fit anywhere, so it was definitely not used.
Maria has exactly 9 white cubes, 9 light-grey cubes and 9 dark-grey cubes, all the same size. She glues them all together to form one larger cube. Which of the cubes below is the one she made?
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Answer: A
Show hints
Hint 1 of 2
A 3x3x3 cube has 27 small cubes; here each colour is used exactly 9 times.
Still stuck? Show hint 2 →
Hint 2 of 2
Count the visible faces of each colour in each option - the right cube must allow exactly 9 of each colour overall.
Show solution
Approach: match the visible colour counts to 9-9-9
The big cube is 3x3x3 = 27 small cubes, painted 9 white, 9 light grey, 9 dark grey.
For each option, see whether the visible and forced hidden cubes can split into three nines.
Only cube A is consistent with using each colour exactly nine times.
Amelia glues six stickers onto the faces of a cube. The figure shows this cube in two different positions. Which sticker is on the face opposite the duck?
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Answer: E
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Hint 1 of 2
Use the two shown views to find which stickers sit next to the duck.
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Hint 2 of 2
Whatever sticker never appears next to the duck in either view sits on the opposite face.
Show solution
Approach: find the duck's neighbours, the rest is opposite
Each cube view shows the duck together with some neighbouring faces.
Collect every sticker seen adjacent to the duck across the two pictures - those four are its side faces.
The remaining sticker, the fly, is opposite the duck.
A square floor is tiled with triangular and square tiles in grey and white. What is the smallest number of grey tiles that must be swapped with white tiles so that the floor looks the same from each of the four marked viewing directions?
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Answer: C — one triangle, one square
Show hints
Hint 1 of 2
Looking the same from all four directions means the pattern must be unchanged by a quarter-turn rotation.
Still stuck? Show hint 2 →
Hint 2 of 2
Find the fewest grey tiles to recolour so every quarter-turn maps grey onto grey.
Show solution
Approach: enforce 4-fold rotational symmetry with fewest swaps
For the floor to look identical from all four sides, the grey pattern must repeat under a 90° rotation.
Compare each tile to where it lands under the rotations and fix the mismatches.
The smallest fix recolours one triangular tile and one square tile.
The first kangaroo is repeatedly mirrored (reflected) across the dotted lines. Two reflections have already been carried out. In which position is the kangaroo in the grey triangle?
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Answer: E
Show hints
Hint 1 of 2
Each step flips the kangaroo across the next dotted edge; reflecting twice restores orientation but moves it.
Still stuck? Show hint 2 →
Hint 2 of 2
Track the kangaroo through the reflections into the grey triangle and read off its pose.
Show solution
Approach: apply successive reflections across the triangle edges
Reflecting across each shared dotted edge flips the kangaroo's orientation in alternating triangles.
Carry the flips along the strip until reaching the grey triangle.
The resulting pose matches option (E).
So the kangaroo in the grey triangle looks like (E).
Clara is forming one big triangle made up of identical little triangles. She has already put some triangles together (see diagram). What is the minimum number of little triangles she still has to add?
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Answer: B — 9
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Hint 1 of 3
A big triangle with little triangles has a square-number count: side 2 holds 4, side 3 holds 9, side 4 holds 16.
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Hint 2 of 3
Find the smallest such big triangle that still fits around the pieces already placed.
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Hint 3 of 3
Then subtract the pieces already there from that total.
Show solution
Approach: complete to the smallest big triangle that fits
The widest row already placed forces the big triangle to be 4 little triangles along each side, and a side-4 triangle holds \(4 \times 4 = 16\) little triangles.
Counting what is already placed and taking it away from 16, Clara must add 9 more little triangles, choice (B).
For the game of chess a new piece, the Kangaroo, has been invented. With each jump the kangaroo jumps either 3 squares vertically and 1 horizontally, or 3 horizontally and 1 vertically, as pictured. What is the smallest number of jumps the kangaroo must make to move from its current position to position A?
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Answer: B — 3
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Hint 1 of 2
Square A is only a short step away, but every jump is long (3 one way, 1 the other), so you must overshoot and come back.
Still stuck? Show hint 2 →
Hint 2 of 2
Check that one or two jumps can never land exactly on A, then find a route that works.
Show solution
Approach: rule out the short jump counts, then exhibit a 3-jump path
Square A sits just one square diagonally from the start, while each jump moves a total of 4 squares (3 + 1), so a single jump lands far away.
Two jumps can be checked to never end exactly on A's square.
Three well-chosen jumps (overshooting and stepping back) do land on A, so the smallest number of jumps is 3.
The faces of a die are labelled 1, 2, 3, 4, 5, 6. Faces 1 and 6 share an edge. So do faces 1 and 5, faces 1 and 2, faces 6 and 5, faces 6 and 4, and faces 6 and 2. Which number is on the face opposite face 4?
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Answer: A — 1
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Hint 1 of 2
Opposite faces of a die never share an edge.
Still stuck? Show hint 2 →
Hint 2 of 2
List every face that face 4 shares an edge with; the one number missing from that list is opposite to 4.
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Approach: find face 4's neighbours; the leftover face is opposite
From the listed common edges, face 4 shares an edge only with face 6.
Face 1 shares edges with 6, 5 and 2, but never with 4, so 1 is not next to 4.
Working through the edges, faces 2, 3, 5 and 6 all end up next to 4, leaving 1 as the only non-neighbour.
A 3×3×3 cube is made of 27 small cubes. Some of the small cubes are removed. Looking at the result from the right, from above, and from the front, you see the same shape each time (shown in the picture). How many small cubes were removed?
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Answer: E — 7
Show hints
Hint 1 of 2
Each of the three views tells you which columns of small cubes are missing in that direction.
Still stuck? Show hint 2 →
Hint 2 of 2
Find an arrangement that produces all three silhouettes at once, then count the empty little-cube spots.
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Approach: match all three silhouettes and count the missing cubes
The three given views show notches: some small cubes must be cleared so the right, top and front outlines all look as drawn.
Removing cubes only from positions that are missing in every relevant view, the fewest consistent removals reproduce all three pictures.
Counting those cleared positions gives 7 little cubes removed.
A 1 × 1 × 1 cube is cut out of each corner of a 3 × 3 × 3 cube. The picture shows the result after the first corner cube has been removed. How many faces does the final shape have?
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Answer: D — 30
Show hints
Hint 1 of 3
Begin with the 6 big faces the cube starts with.
Still stuck? Show hint 2 →
Hint 2 of 3
Look at the one notch in the picture: it scoops out a little corner and reveals 3 new small square walls.
Still stuck? Show hint 3 →
Hint 3 of 3
There are 8 corners, so add up all the new little faces and the original 6.
Show solution
Approach: count original faces plus faces added per corner
Even after the corners are scooped out, each of the 6 big outer faces is still one face (just with bites taken out of it), so that is 6 faces.
Each corner cut opens up 3 new little square faces inside the notch, and there are 8 corners: \(8 \times 3 = 24\) new faces.
There are numbers on the middle part of a 3-part unfolded card. The left and right parts of the card have holes. Mike folds the right part along the dotted line onto the middle part. He can now see the numbers 2, 3, 5 and 6 through the holes. Then he folds the left part along the dotted line onto the other two parts. What is the sum of the numbers that he can still see through the holes?
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Answer: A — 8
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Hint 1 of 2
After folding the right flap, you already see 2, 3, 5 and 6 through its holes.
Still stuck? Show hint 2 →
Hint 2 of 2
Folding the left flap on top covers some of those holes; only the numbers under a left-flap hole stay visible.
Show solution
Approach: trace which holes still line up after both folds
After the right flap is folded over, its holes already let Mike see 2, 3, 5 and 6 on the middle panel.
When the left flap folds on top, its holes only line up over some of those numbers: two of them stay showing through a hole and the other two get covered by solid paper.
The two numbers still visible through a hole are 3 and 5, so the sum is 3 + 5 = 8, giving the answer (A) 8.
Which of the five shapes cannot be placed on the large square so that it only lies on white squares? (The five shapes A–E and the patterned large square are pictured with the question.)
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Answer: D
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Hint 1 of 2
Look at where the white squares actually sit on the big board, then try to slide each shape around so all of its squares land on white.
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Hint 2 of 2
Four of the shapes can be tucked onto a run of white squares; hunt for the one shape whose squares are forced to grab a black square no matter where you put it.
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Approach: try to fit each shape onto only white squares
A shape works only if you can lay it down so every one of its squares sits on a white square of the board.
Slide each shape A–E around the board: four of them can be placed on a stretch of white squares with no black square underneath.
Shape D is the only one that always lands on at least one black square wherever it goes, so it cannot sit only on white squares — the answer is (D).
A dark disc with three holes is placed on top of the dial of a watch (see picture). The disc is then rotated about its centre. Which three numbers can be seen through the holes at the same time?
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Answer: A — 4, 6 and 12
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Hint 1 of 2
Rotating the disc keeps the three holes the same distances apart around the circle.
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Hint 2 of 2
Compare the gaps between the three holes with the gaps between the numbers in each answer; only a matching gap pattern can appear.
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Approach: match the angular gaps of the three holes to the gaps between the numbers
The three holes sit at fixed clock-positions, so the gaps between them (measured in hours) stay the same no matter how you turn the disc: the gaps are 2, 4 and 6 hours.
Check the gaps for each answer: 4, 6 and 12 are spaced 2, 4 and 6 hours apart — the same pattern.
None of the other answers has gaps 2, 4, 6, so only this triple can show through the holes at once.
All vehicles in the garage can only drive forwards or backwards. The black car wants to leave the garage (see diagram). What is the minimum number of grey vehicles that need to move at least a little bit so that this is possible?
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Answer: C — 4
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Hint 1 of 3
Find the exit first, then the straight lane the black car must drive along to reach it.
Still stuck? Show hint 2 →
Hint 2 of 3
Only the grey vehicles actually sitting in that lane (or blocking a vehicle that does) need to move.
Still stuck? Show hint 3 →
Hint 3 of 3
Count just those blockers - vehicles parked out of the way can stay put.
Show solution
Approach: clear the black car's exit lane, moving only the blockers
The black car must drive straight to the opening on the right.
Identify every grey vehicle sitting in or across that path.
Exactly 4 of them must shift at least a little to free the route.
Marc builds the number 2022 from 66 cubes of the same size, all glued together (see picture). He then paints the entire outer surface. On how many of the 66 cubes has Marc painted exactly four faces?
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Answer: E — 60
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Hint 1 of 2
Every face glued to a neighbouring cube is hidden; all the rest get painted.
Still stuck? Show hint 2 →
Hint 2 of 2
Since the digits are one cube thick, a cube shows 4 painted faces exactly when it touches 2 neighbours — so look for the few cubes that touch only 1.
Show solution
Approach: count cubes that touch exactly two neighbours
The digits are one cube thick, so every cube always shows its front and back; it shows exactly 4 painted faces when it also has exactly 2 of its in-plane neighbours, i.e. it sits in a straight run or at a corner.
The 0 is a closed loop, so every one of its cubes has 2 neighbours and shows 4 faces. Each 2 is an open strip with exactly two free ends, and those end cubes have only 1 neighbour, so they show 5 painted faces.
The only exceptions are the 2 free ends on each of the three 2s, which is 6 cubes in all.
That leaves 66 − 6 = 60 cubes with exactly four painted faces, so the answer is E.
A triangular pyramid is built with 10 identical balls. Each ball has one of the letters A, B, C, D and E on it, and there are 2 balls marked with each letter. The picture shows 3 side views of the pyramid. What is the letter on the ball with the question mark?
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Answer: A — A
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Hint 1 of 2
Ten balls, two of each of A–E; the three side views show different faces of the same pyramid.
Still stuck? Show hint 2 →
Hint 2 of 2
Use the visible letters to deduce which letters are already placed, leaving the '?' ball's identity.
Show solution
Approach: reconcile the three views
Each letter appears exactly twice among the ten balls, and the three views show the pyramid from different sides.
Tracking which positions carry which letters across the views fixes every ball except the marked one.
Bridget folds a square piece of paper in half, then in half again, and then cuts it along the two lines shown in the picture. How many pieces of paper does she get?
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Answer: C — 9
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Hint 1 of 3
Folding the square in half twice stacks it into four layers.
Still stuck? Show hint 2 →
Hint 2 of 3
One snip through four layers makes four cuts at once, so imagine the cut lines reflected when you unfold.
Still stuck? Show hint 3 →
Hint 3 of 3
Draw the unfolded square with all the cut lines and count the separate pieces.
Show solution
Approach: track the cuts through the folded layers, then unfold
Folding the square twice stacks it into four layers.
The two cuts slice through all the layers; unfolding turns each cut into a full line across the paper.
Counting the regions those lines make gives 9 separate pieces (C).
The given net is folded along the dotted lines to form an open box. The box is placed on the table so that the opening is on top. Which side is facing the table?
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Answer: B — B
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Hint 1 of 2
Fold the net up in your head into an open box (one face missing for the opening).
Still stuck? Show hint 2 →
Hint 2 of 2
With the opening on top, the face opposite the opening is the one on the table.
Show solution
Approach: fold the net up and find the bottom face
Fold the net up so the four sides stand and one face is missing; that missing face is the opening on top.
The face that lies flat at the bottom, opposite the opening, is the one touching the table, which is B, choice (B).
A magnified circular patch must match a piece of the squiggly drawing exactly in how the lines cross it.
Still stuck? Show hint 2 →
Hint 2 of 2
Compare the line pattern inside each circle with what actually appears in the picture; one pattern never occurs.
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Approach: match each magnified circle to a region of the picture
Through the magnifying glass Peter sees a round window onto part of the drawing, so the lines inside the circle must reproduce a real crossing in the picture.
Four of the circles match a place where the curves cross or pass through as shown.
The pattern of lines in circle E does not occur anywhere in the picture, so that is the section he cannot see.
Each piece is a square with some sides pushed in (a dent) or pushed out (a bulge).
Still stuck? Show hint 2 →
Hint 2 of 2
When two pieces sit side by side, a bulge on one must drop into a matching dent on its neighbour; find the piece whose curves have no partner.
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Approach: match each piece's curved edges so bulges fill dents
To build a square with straight outer sides, every outward bulge on one piece must fit a matching inward dent on a neighbour, so the curved edges have to pair up.
Four of the pieces have curves that pair off neatly and tile a 2-by-2 square.
Piece B's curves cannot be matched by the others, so it is the piece left over.
Anne has several grey tiles shaped like the one in the picture. What is the greatest number of these tiles she can place on the 5 × 4 rectangle without any overlaps?
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Answer: C — 4
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Hint 1 of 3
First count how many squares the board has and how many each tile covers.
Still stuck? Show hint 2 →
Hint 2 of 3
Area says at most 5 tiles could fit, but try actually drawing them in.
Still stuck? Show hint 3 →
Hint 3 of 3
The bumpy T-shape always leaves a few squares stranded, so you can't reach 5.
Show solution
Approach: bound by area, then test placement
The board has \(5 \times 4 = 20\) squares and each grey tile covers 4 squares, so at most \(20 \div 4 = 5\) tiles could fit.
But when you slot the T-shaped tiles in, they keep leaving small gaps, so 5 is impossible.
You can fit 4 tiles with no overlap (covering 16 squares), so the most is 4, choice C.
Patricia drives one afternoon at a steady speed to her friend. She looks at her watch when she leaves and again when she arrives (both clocks are shown). Where will the minute hand be when she has completed one third of her journey?
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Answer: D
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Hint 1 of 3
The two clocks show the start time and the finish time of the whole drive.
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Hint 2 of 3
Find how far the minute hand swings between those two clocks, then take one third of that swing.
Still stuck? Show hint 3 →
Hint 3 of 3
Mark the one-third point and match it to the pictured clock faces.
Show solution
Approach: take one‑third of the way between start and finish
From the leaving clock to the arriving clock, the minute hand sweeps through a fixed amount.
Since the speed is constant, one third of the journey means the hand has swept one third of that amount.
Marking the one-third point of the swing matches the clock in choice D.
Johann stacks \(1\times1\) cubes on the squares of a \(4\times4\) grid. The diagram shows how many cubes are piled on each square. What will Johann see if he looks at the tower from behind?
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Answer: C
Show hints
Hint 1 of 3
Looking from the back flips left and right compared with the front.
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Hint 2 of 3
For each line of squares, only the tallest stack shows up in the side view.
Still stuck? Show hint 3 →
Hint 3 of 3
Read off the tallest stack in each row, then flip the row left-to-right for the back view.
Show solution
Approach: read the height grid from the back view
The grid tells how tall each stack is; from behind you see the same stacks but with left and right swapped.
For each line going across, the tallest stack is the one that shows in the outline.
Reading the tallest stacks and flipping left-to-right gives the shape in choice C.
Lisa built a large cube out of 8 smaller ones. The small cubes have the same letter on each of their faces (A, B, C or D). Two cubes with a common face always have a different letter on them. Which letter is on the cube that cannot be seen in the picture?
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Answer: B — B
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Hint 1 of 3
There are 8 little cubes but only 4 letters, and every cube touches three neighbours that must all differ from it.
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Hint 2 of 3
Two cubes can share a letter only if they do NOT touch, i.e. they sit at opposite ends of a long diagonal through the centre.
Still stuck? Show hint 3 →
Hint 3 of 3
The hidden cube is the corner diagonally opposite a visible one, so it copies that cube's letter.
Show solution
Approach: opposite corners share a letter
Each small cube touches 3 others (one in each direction), and touching cubes must differ, so a cube and its 3 neighbours use up all 4 letters A, B, C, D.
That means a letter can repeat only on two cubes that never touch, namely the two ends of a diagonal running through the centre of the big cube.
So each of the 4 space-diagonals carries one repeated letter, pairing every cube with the corner diagonally across from it.
The unseen back corner is diagonally opposite a visible corner, and matching their letter gives the hidden one: B.
Fridolin the hamster runs through the maze in the picture. 16 pumpkin seeds are lying on the path. He is only allowed to cross each junction once. What is the maximum number of pumpkin seeds that he can collect?
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Answer: B — 13
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Hint 1 of 2
He may pass each junction only once, so he can't take every seed.
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Hint 2 of 2
Find the longest single path through the maze and count seeds along it.
Show solution
Approach: find the best non-repeating path
Fridolin must follow a path that uses each junction at most once.
Because some seeds sit on junctions he cannot revisit, he cannot collect all 16.
The best possible single path lets him pick up 13 of the pumpkin seeds.
In the box there are seven blocks. By sliding the blocks around it is possible to make room so that one more block can be added. What is the least number of blocks that must be moved?
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Answer: B — 2
Show hints
Hint 1 of 2
Look at where the empty space is — it is split into pieces, not one block-sized hole yet.
Still stuck? Show hint 2 →
Hint 2 of 2
You only need to slide enough blocks to gather that empty space into one spot the new block fits.
Show solution
Approach: gather the empty space into one hole
There is exactly one block of empty space, but it is spread out, so a new block will not fit yet.
By sliding just two of the blocks, the scattered empty space lines up into a single block-shaped gap.
Lines are drawn on a piece of paper and some of the lines are numbered. The paper is cut along some of these lines and then folded into the shape shown. Along which lines were the cuts made?
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Answer: B — 2, 4, 6, 8
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Hint 1 of 2
A fold keeps the paper joined, but a cut lets a flap lift up and stand free.
Still stuck? Show hint 2 →
Hint 2 of 2
Match each free-standing flap in the folded picture back to its numbered line on the flat sheet.
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Approach: unfold the model in your head
On the flat sheet, a fold-line stays attached but a cut-line frees a flap to be raised.
Tracing the flaps that lift up in the folded picture back to the sheet, they sit on the even-numbered lines.
So the cuts were made along lines 2, 4, 6 and 8 (answer B).
Joanna divides the figure into five equal-sized, same-shaped parts, each of which consists of three squares. Which of the letters is in the part with the star?
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Answer: E — E
Show hints
Hint 1 of 2
All five pieces are the same shape made of three squares, so figure out that shape first from a corner that can only be filled one way.
Still stuck? Show hint 2 →
Hint 2 of 2
Once you know the piece shape, build outward and watch which three squares end up grouped with the star.
Show solution
Approach: find the repeating 3-square piece, then read off the star’s group
Since every piece is the same three-square shape, start at a corner of the figure where only one shape can fit; that fixes what the repeating piece looks like.
Lay that same piece again and again to tile the whole figure with no gaps or overlaps — there is only one way it all fits together.
The piece that ends up covering the starred square also covers the square labelled E, so the answer is (E).
Julio wants to make the shape shown in the top picture on the right. He has several of each of the five tiles shown in the bottom picture on the right. The tiles must be placed next to each other without overlapping. What is the smallest number of tiles he must use?
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Answer: C — 13
Show hints
Hint 1 of 2
To use as few tiles as possible, you want each tile to cover as much of the cross as it can, so reach for the biggest tiles first.
Still stuck? Show hint 2 →
Hint 2 of 2
The straight parts of the cross are easy to cover with the large rectangle and big triangle; the pointy arm-tips are what force you to use the small triangles.
Show solution
Approach: cover the big areas with big tiles, the tips with small ones
Fewer tiles means each tile should cover as much as possible, so fill the wide straight parts of the cross with the largest tiles (the long rectangle and the big triangle).
The four slanted arm-tips are too thin for the big tiles, so each tip has to be finished with the small triangle pieces — these are unavoidable and set the limit on how low the count can go.
Packing the big tiles in the body and the small triangles at the tips, with no overlaps, covers the whole cross in 13 tiles, and no arrangement does it in fewer, so the answer is (C) 13.
Tina wants to combine the three building blocks shown in the picture to form a cube building. Which one of the following cube buildings could she make? (The three blocks and the five choices A–E are pictured with the question.)
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Answer: D
Show hints
Hint 1 of 2
First just count: how many little cubes are in the three blocks all together? The answer building must use exactly that many cubes.
Still stuck? Show hint 2 →
Hint 2 of 2
Throw out any choice with the wrong cube count, then check the survivors by mentally snapping the three blocks together.
Show solution
Approach: count cubes first, then fit the blocks
Each of the three building blocks is made of small cubes; counting them gives a fixed total number of cubes that the finished building must contain.
Count the cubes in each answer building and cross out the ones with the wrong total — the right building must have exactly as many cubes as the three blocks combined.
Among the buildings with the correct cube count, only (D) can actually be assembled from those three particular blocks fitting together with no gaps, so the answer is (D).
A building is made of cubes of the same size. The three pictures show it from above (von oben), from the front (von vorne) and from the right (von rechts). What is the maximum number of cubes that could be used to make this building?
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Answer: B — 19
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Hint 1 of 2
The top view fixes which columns can hold cubes; the front and side views cap each column's height.
Still stuck? Show hint 2 →
Hint 2 of 2
For the maximum, make every column as tall as its views allow.
Show solution
Approach: raise each column to the height its views permit
The top view shows which floor positions are occupied.
The front and right views give the largest height allowed for each row and column.
Stacking each column to its maximum allowed height totals 19 cubes.
Maria pours 4 litres of water into vase I, 3 litres into vase II and 4 litres into vase III, as shown. Seen from the front, the three vases look the same size. Which of the following pictures can show the three vases seen from above?
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Answer: A
Show hints
Hint 1 of 2
Same water heights from the front but different amounts means the vases have different base areas.
Still stuck? Show hint 2 →
Hint 2 of 2
Vase II holds less (3 L vs 4 L) at the same height, so II has the smaller top - match the top-view sizes.
Show solution
Approach: use volume = base area x height to rank the tops
From the front the vases look the same size, so the shown heights reflect base area, not real width.
Vases I and III hold 4 L and II holds 3 L; with the heights shown, the top-view areas differ accordingly.
The top view giving I and III equal larger tops and II a smaller top is option A.
John built a structure of equal-sized wooden cubes whose front, right-side and top views are shown, using as many cubes as possible. His sister Ana wants to remove as many cubes as she can without changing any of these three views. At most, how many cubes can she remove?
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Answer: B — 12
Show hints
Hint 1 of 2
The three views (front, side, above) must all stay the same after removing cubes.
Still stuck? Show hint 2 →
Hint 2 of 2
Keep only the cubes forced by all three silhouettes; count how many of the fullest build can be taken away.
Show solution
Approach: compare the fullest build with the minimum that keeps all views
Build the most cubes giving those three views, then strip out any cube not needed by all three silhouettes.
Each removed cube must leave the front, side and top outlines unchanged.
The largest number she can remove while preserving every view is 12.
Dirce built the sculpture shown by gluing together cubic boxes that are half a metre on each side. She then painted the whole sculpture except the base it rests on, using a special paint sold in cans. Each can covers 4 square metres. How many cans of paint did she have to buy?
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Answer: B — 4
Show hints
Hint 1 of 2
Each cube edge is 0.5 m, so a small face is 0.25 m^2; count painted faces of the stepped solid, skipping the base.
Still stuck? Show hint 2 →
Hint 2 of 2
Total painted area / 4 m^2 per can, then round up to whole cans.
Show solution
Approach: count exposed faces, convert to area, divide by can coverage
Each cube is 0.5 m on a side, so one face is 0.5x0.5 = 0.25 m^2.
Count every exposed face of the stepped solid except the bottom support; multiplying by 0.25 gives the painted area.
Dividing by 4 m^2 per can and rounding up, she needs 4 cans.
Vania has a sheet of paper divided into nine equal squares. She folds it as shown — first the horizontal folds, then the vertical folds — until the coloured square is on top of the stack. She wants to write the numbers 1 to 9, one per square, so that after folding they read in order from top to bottom, starting with 1 on top. On the unfolded sheet shown, which numbers should she write in places a, b and c?
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Answer: C — a = 7, b = 5, c = 3
Show hints
Hint 1 of 2
Track where each unfolded square ends up in the stack after the horizontal then vertical folds.
Still stuck? Show hint 2 →
Hint 2 of 2
Reverse the folds to read which numbers land at positions a, b and c on the flat sheet.
Show solution
Approach: reverse the fold order to map stack layers to grid cells
Folding horizontally then vertically stacks the nine squares; the coloured square is on top (number 1), and lower layers get 2,3,...
Unfolding to the flat sheet, each layer returns to its cell, spreading the numbers in a fixed pattern.
Reading positions a, b, c gives a = 7, b = 5, c = 3 - option C.
Nina wants to make a cube from the paper net. You can see there are 7 squares instead of 6. Which square(s) can she remove from the net, so that the other 6 squares remain connected and from the newly formed net a cube can be made?
Show answer
Answer: D — only 3 or 7
Show hints
Hint 1 of 2
A cube net needs the remaining 6 squares to stay connected AND to fold up without two squares landing on the same face.
Still stuck? Show hint 2 →
Hint 2 of 2
Test each candidate removal: most leave a shape that overlaps when folded; only certain end squares work.
Show solution
Approach: test which removals leave a connected, foldable 6-square net
Removing a square must keep the other six joined and able to fold into a cube with no doubled-up face.
Taking out an interior square breaks the net or makes two squares fold onto the same face, so those fail.
Removing square 3 works, and removing square 7 works, while no other single removal does — so the answer is only 3 or 7.
Maria writes a number on each face of the cube. Then, for each corner point of the cube, she adds the numbers on the faces which meet at that corner. (For corner B she adds the numbers on faces BCDA, BAEF and BFGC.) In this way she gets a total of 14 for corner C, 16 for corner D, and 24 for corner E. Which total does she get for corner F?
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Answer: C — 22
Show hints
Hint 1 of 2
Two corners at opposite ends of a space diagonal use all six faces between them, so their corner-sums add up to the same total every time.
Still stuck? Show hint 2 →
Hint 2 of 2
Pair the given corner with the unknown one along a space diagonal, and pair the other two the same way.
Show solution
Approach: use that opposite corners of the cube share the same total of all six faces
A corner's number is the sum of its three meeting faces. Two corners on opposite ends of a space diagonal together touch all six faces exactly once, so each such pair has the same sum S (the total of all six faces).
Corners C and E are opposite, and corners D and F are opposite, so C + E = D + F.
A model train set has only identical curved track pieces. Matthias uses 8 of them to make a closed circle (picture on the left). Martin starts his track with the 2 pieces shown on the right. He also wants a closed track using as few pieces as possible. How many pieces will his track use?
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Answer: B — 12
Show hints
Hint 1 of 3
Eight identical pieces close a full circle, so each piece bends the track by \(360^\circ \div 8 = 45^\circ\).
Still stuck? Show hint 2 →
Hint 2 of 3
To come back to the start, all the bends together must add up to one full turn, \(360^\circ\).
Still stuck? Show hint 3 →
Hint 3 of 3
Martin's two starting pieces curve opposite ways and cancel, so count how many more pieces are needed to still make a full turn.
Show solution
Approach: use the fixed bend angle to close a loop
Each curved piece turns the track \(45^\circ\), because 8 of them make a full circle (\(8 \times 45^\circ = 360^\circ\)).
To form a closed track the bends must add up to a full turn of \(360^\circ\), but Martin's two opening pieces curve in opposite directions and cancel out.
Working out the smallest loop that still turns a full \(360^\circ\) starting from that S-shape needs 12 pieces in all, which is choice B.
