Spatial and visual: open practice pack: worked explanations

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Rotate one arm at a time rather than the whole shape at once. A clockwise quarter turn sends anything pointing up to pointing right, so the tall arm lies down along the top; it sends anything pointing right to pointing down, so the foot now hangs downwards. Both arms still meet at the same corner, which is now the top-left. Putting the bar along the bottom is what a quarter turn the other way produces, and leaving the upright on the right is a mirror image rather than a rotation.

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A reflection reverses the axis that crosses the mirror line and leaves the axis running along it untouched. This mirror line runs north to south, so east and west swap while north and south stay put: the desk keeps its north wall but slides to the west corner, and the door moves to the middle of the east wall. Moving the desk to the south wall means you reflected across an east-west line instead, and moving only one of the two items means you stopped halfway.

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Count a solid in groups rather than one edge at a time, so nothing is missed or double counted. There are four edges around the square base, and four sloping edges running from the base corners up to the apex, giving four plus four. The count of five belongs to the faces, not the edges: one square plus four triangles. Twelve is the edge count of a box, which has an extra square face where this solid has only a point.

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Read a small matrix in two directions: across for what changes, down for what stays. Moving left to right, the outline is unchanged and a dot appears inside it, so the bottom row must keep its square outline and gain a dot in the same place. Keeping the circle means you copied the row above instead of applying its rule, and shading the whole square swaps in a different change from the one the top row demonstrates.

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Count by size, largest group first, and the total is hard to get wrong. There are six single cells. A two-by-two square needs two adjacent columns, and with three columns there are only two such positions, so two larger squares exist, giving six plus two. Stopping at six means you counted only the single cells; nine comes from assuming there is a two-by-two square starting at each of the three columns, but the one starting at the third column would run off the edge.

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A half turn moves every point to the position directly opposite it through the centre of the card, which means both directions reverse at once: left becomes right and top becomes bottom. The top-left notch therefore lands at the bottom-right, and a hole that was low and right of centre becomes high and left of centre. Reversing only the up-down sense is a flip over a horizontal line, not a turn, and moving the notch while leaving the hole low is the commonest slip. Every feature has to travel.

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Fold the row first and the extras second. A row of four wraps into a closed band of four side faces, which leaves exactly two squares to supply the lid and the base, so those two must hang off different faces of the band and on opposite sides of it. Two squares attached above neighbouring positions both try to become the lid and collide, and a row of five wraps too far, so its fifth square lands on top of the square it started from. A solid block of two rows of three has no row long enough to make a band at all, so it cannot close either end.

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Count a stack layer by layer, turning each layer's description into a number of cubes before adding anything. A three-by-three layer is nine cubes, a two-by-two layer is four, and the cap is one, so the total is nine plus four plus one. Adding the side lengths instead of the areas gives three plus two plus one, which counts rows rather than cubes; stopping at thirteen means the single cube on top was forgotten; and nine is the base alone, ignoring everything you cannot walk around.

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Reflecting in the leading diagonal simply swaps each cell's row number with its column number, because that diagonal is the line where the two are equal. Row one, column two becomes row two, column one; row three, column one becomes row one, column three. Reflecting in a vertical line instead would leave both counters in their own rows, and a half turn would send the top counter to the bottom row rather than to the middle row.

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When a matrix has more than one feature, decide which feature is fixed and which one moves before predicting anything. The mark stays the same all the way along a row, while the count rises by one at each step. The upper row settles the question of what the count does: one, two, three is a steady increase rather than a doubling, so a lower row that reads one, two must continue with three of the same mark. Doubling would give four and only looks plausible if you ignore the third box above.

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Test a shape for line symmetry by looking for a fold that sends each corner onto another corner of the same kind. A triangle with two equal sides folds along the line through the point where those sides meet; a rectangle folds along either line joining the midpoints of opposite sides; a semicircle folds along the line through the middle of its straight edge. A right-angled triangle with unequal shorter sides has three corners that are all different, so no fold can pair them, and folding along the longest side merely doubles it rather than matching halves.

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Treat the crease as a mirror and count layers first. The punch passes through two layers, so unfolding must produce two holes, and the second one is the mirror image of the first in the crease line. The crease here runs horizontally across the middle, so the pair sit at equal distances above and below it while their distance from the left edge is untouched. Putting one hole in each half means you mirrored in a vertical line, which is what a left-to-right fold would have created.

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Split a compound pattern into separate tracks and continue each one on its own. The side count climbs by one at every step, so a row reading four sides then five sides must reach six. The shading alternates plain, shaded, plain, so a step that follows a shaded figure returns to plain. Keeping the shading on is the usual slip, because the eye latches onto the most recent figure instead of the alternation, and repeating five sides ignores the count track altogether.

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When two features change together, work out how long each one takes to return to its starting state and then use the remainder. Direction repeats every four moves, so three clockwise quarter turns from up gives right, then down, then left. Shading repeats every two moves, and three is an odd number of swaps, so it ends reversed: the head is plain and the tail shaded. Keeping the head shaded treats three swaps as if they cancelled, and stopping at down means only two moves were counted.

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Moving to the opposite side of a line of objects does not move the objects; it swaps which end of the line is on your left. The order along the ground is fixed, so the second viewer reads exactly the same list backwards, and the post at each end trades places with the post at the other end. Swapping only the outer pair treats the change as a reflection applied to the ends alone, and leaving the list unchanged forgets that your left hand now points the other way along the line.

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Stop counting shapes and start counting lines. A rectangle is fixed the moment you choose two of the vertical grid lines for its sides and two of the horizontal grid lines for its top and bottom. There are five verticals, which can be paired in ten ways, and three horizontals, which can be paired in three ways, so ten times three gives the total. Counting only the single cells gives eight, counting only the shapes that are squares gives eleven, and stopping at twenty means the full-height rectangles were never paired with the shorter ones.

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A flap turns about the edge where it meets the base, so ask how each direction drawn on the flap relates to that hinge. Directions running parallel to the hinge are unaffected by the fold, while directions running away from the hinge swing through a quarter turn and end up vertical. The arrow runs directly away from the base, which is directly away from the hinge, so it finishes pointing upwards. Pointing across the box would need the wall to fold past upright and lie flat again, and pointing down would need it to fold the other way, inwards over the base.

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Count layers, then undo the folds in reverse order, mirroring the whole set of holes in each crease as you go. Two folds make four layers, so a single punch must give four holes. Undoing the vertical crease mirrors the punch left to right; undoing the horizontal crease then mirrors both of those up and down, and because the punch sat at the centre of the folded quarter, the four holes land at the centre of each quarter of the whole sheet. Finding only two holes is the sign that one of the two folds was never undone.

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This is elimination on two independent features, exactly as you would fill the last cell of a completed table. The bottom row already holds a circle and a triangle, so the missing shape is a square; the same row already holds two dots and three dots, so the missing count is one. Checking down the last column agrees: it holds a circle and a triangle, and counts of three and two. Taking the count from the cell directly above gives a square with three dots, but three has already been used in that column, which is how you catch the error.

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Take one movement at a time and write out all six positions after each. Tipping forward cycles four faces, top to front to bottom to back to top, and leaves left and right alone, so after the tip the square from the far face is on top and the star faces you. A quarter turn clockwise from above cycles the four side faces, front to left to back to right to front, and leaves top and bottom alone, so the arrow swings round from the right to the front while the square stays on top. Turning the other way would bring the triangle forward instead.

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Add the lines one at a time and count how many existing regions each new line actually passes through, because a line adds exactly that many regions. The first diagonal makes two regions; the second crosses it and passes through two of them, making four. The third line runs through the same crossing point at the centre, so it passes through only the left and right regions and adds two more, not four. Expecting eight assumes the third line meets the other two at two separate points, which concurrency at the centre prevents.

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Keep two columns rather than drawing the path: one for the heading, one for the running totals east and north. Update the heading first at every turn, remembering that a right turn runs north to east to south to west, and that when you face south your left hand points east. The legs then total three north, two east, three south, one east, two north, which nets to three east and two north, and the final turn leaves you facing north again. Reading the totals off in the wrong order is what produces two east and three north.

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Undo the folds in reverse order and mirror the holes in each crease, using the crease line rather than the middle of the sheet. The packet's left edge sits three units in, so a punch two units into it is five units from the left edge of the whole sheet; the packet's foot is the mid-height crease, so a punch two units above it is six units up. Undoing the halving fold mirrors that hole about mid-height, giving a second at a height of two. Undoing the off-centre fold mirrors both about the line three units in, and five is two units past that crease, so its partner is one unit from the edge. Assuming the vertical crease was central gives three and five instead, which is the slip this puzzle is built to catch.

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When a matrix resists a single reading, test each feature against a different direction. Shape and dot count behave by row: a row keeps one shape throughout and adds a dot at each step, and each row begins one dot higher than the row above, so the bottom row runs three, four, five circles. Shading obeys no row rule at all, and following it cell by cell shows it stepping one place to the right as it moves down, which puts it on the diagonal and therefore on the bottom-right cell. Treating shading as a row feature is what leaves the cell plain.

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Rotations and reflections do not commute, so apply them strictly in the order given and track two things: where the pole's top end goes, and which side of the pole the flag sits on. The clockwise quarter turn sends up to right and right to down, so the pole lies flat with its top end to the right and the flag hangs below it. The vertical mirror then swaps left and right, moving the top end to the left and carrying the flag with it, while below stays below. Doing the mirror first would put the flag on the left, and the same turn would then send left to up, which is why the flag would stand up instead of hang, the tell-tale sign that the order matters.