Shape Transformation, Folding and Pattern Recognition

Paper folding and cut-out patterns

Paper folding and cut-out patterns test whether you can predict the shape of a sheet after it is folded, cut, and then unfolded. The central rule is that a fold is a reflection. Every point on one side of a crease moves to the mirror position on the other side, while points lying on the crease stay where they are. If you trace points through each fold instead of guessing the final picture, most questions in this topic become routine.

• crease (fold line) — the line along which paper is bent. Points on the crease do not move. Creases are the only places where the sheet is mirrored.
• superimpose — to lay one layer over another so that their marks overlap. On a transparent sheet both layers stay visible, so marks from the folded part and the unfolded part appear together. Example: a dot on the right of a crease, reflected to the left, appears on the left side of the folded figure.
• mirror image — the figure obtained by reflecting a shape across a line, so that each point is the same distance from the line as its image, on the opposite side.
• punched (cut-out) — a hole or notch cut while the paper is still folded. After unfolding, the same cut appears at every mirror position.
• layer — one thickness of paper in a folded stack. Each fold in half doubles the number of layers.

When a transparent sheet is folded along a dotted line, the folded figure is the union of the part that stays in place and the mirror image of the part that crosses the line. Marks on the crease itself stay unchanged. A reflected arm of a V-shape or < shape meets its original arm on the fold line, so the result often looks like a chevron or zigzag with its vertices on the crease. Each mark in the answer must therefore come from one of these two sources, and any option that places a mark on the far side of the crease where neither source has one can be rejected.

When a folded sheet with cut-outs is unfolded in reverse order, each cut is copied to its mirror position across every crease used. A cut that touches a crease unfolds into a shape symmetric about that crease, extended on both sides and twice as wide measured perpendicular to the crease. With two perpendicular creases, each cut produces four copies related by reflection across both lines, and a cut centred on the intersection of the creases stays centred. Diagonal creases work the same way, with reflections across both diagonals.

Folding a rectangle in half along a line parallel to one side halves the side perpendicular to that line, while the other side is unchanged. The perimeter of a rectangle with sides a and b is P = 2(a+b). Joining the short edges of the current shape creates a crease parallel to them, which halves the long side. Joining the long edges halves the short side. Because the longer side may change after a fold, you must recompute which side is longer before each step. For example, a 40 \times 10 cm sheet folded along a crease parallel to its short edges becomes 20 \times 10 cm, with perimeter 2(20+10) = 60 cm. The same method gives the final perimeter of any sequence of halvings.

Each fold in half doubles the number of layers and halves the area of each layer. After n folds of this kind the stack has 2^n layers, and each layer covers \frac{1}{2^n} of the original area when all layers coincide. A diagonal fold of a square gives a right isosceles triangle of half the area, and folding that triangle along its own line of symmetry continues the same doubling.

When a sheet is folded in half n times in the same direction, the k-th fold creates 2^{k-1} creases, so the total number of creases in the unfolded sheet is 1 + 2 + \cdots + 2^{n-1} = 2^n - 1. The creases are evenly spaced, at multiples of \frac{L}{2^n} along a sheet of length L.

For a die, several views show which faces meet at a common corner. Two faces that never appear together at a corner in any view must be opposite. Every face of a cube is adjacent to four others and opposite to exactly one. In a net, a straight row of four squares places the first and third squares in opposite positions, and the second and fourth in opposite positions. Two faces that share an edge in the net cannot be opposite on the cube. When two candidate nets give the same opposite pairs, compare the cyclic order of the three faces around a corner. A mirror-reversed arrangement cannot fold into the same die.

To count the folds needed to form a closed solid from a flat sheet, count the creases along which the sheet must bend to form the edges of the solid. Boundaries of the net that are cut edges are not bent and are not counted. A crease that lies flat in the final object is not a fold.

Figure pattern recognition

Figure pattern recognition asks you to find the single rule that links a series of figures or tiles, and then to apply that rule to choose the missing or next figure. Most items reduce to a few rule families, so the first step is to decide which attribute changes from one figure to the next, such as a count, an angle, a position, an orientation, or a shading. Once a candidate rule is found, test it on every given figure, not only on the first two, before accepting it.

Counting rules. When each figure carries a number of dots, sides, or shaded cells, the count usually follows a named integer sequence. The consecutive primes are 2, 3, 5, 7, 11, 13, so a missing count lying between 3 and 7 must be 5, and you should check the prime list rather than guess . The Fibonacci sequence 0, 1, 1, 2, 3, 5, 8, 13 is built by adding the two preceding terms, so after counts of 3 and 5 the next count must be 8 . A polygon with n sides may also be linked to n!, so a triangle gives 3! = 6, a quadrilateral gives 4! = 24, a pentagon gives 5! = 120, and a hexagon gives 6! = 720 .

Equation-style rules. Some items hide an equation whose left side is a difference of two squares. The identity a^2 - b^2 = (a-b)(a+b) reduces the left side to a linear expression when a and b differ by a constant. The resulting equation can be solved and then verified by direct substitution .

Angular and fractional progressions. When a shaded sector grows, its angle usually increases by a fixed step. Angles of 90^\circ, 135^\circ, and 180^\circ rise by 45^\circ = \tfrac{1}{8} of a full turn, so the next angle is 225^\circ = \tfrac{5}{8} of the circle . A fraction that grows by a fixed step must be converted to a common form before comparing, and the rule can be checked in either form . The starting edge of a sector may advance by a fixed angle, such as 90^\circ per step, and clock positions make this easy to track, for example 9, 12, 3, then 6 o'clock .

Rotation and reflection.

• Clockwise 90^\circ rotation — turns every part a quarter-turn clockwise about the axis perpendicular to the plane, so a mark at the upper right moves to the lower right.
• Reflection along a horizontal line — exchanges the top and bottom halves, while a reflection along a vertical line exchanges left and right.
• 180^\circ rotation — moves a mark at the upper right to the lower left.
• 45^\circ rotation — turns an axis-aligned square lattice into a diamond while keeping the relative arrangement of the blocks.

A composite transformation is identified by applying each candidate operation in order and checking the resulting position of a distinctive part . A 45^\circ clockwise rotation must preserve arm directions and block order, so a mirror image or a rotation in the opposite sense is wrong .

Cyclic positional shifts. When a marked cell moves one row down at each step and a cell leaving the bottom row returns at the top, the row index after k steps in an n-row grid is r' = ((r - 1 + k) \bmod n) + 1, while the column stays fixed .

Line totals. When the count of filled cells is constant along each row and each column, the missing count in a line equals that constant minus the known counts in the same line .

Repeated motifs. A large figure built by repeating one unit is generated by the pattern whose exact arrangement of blocks and arms reproduces that unit. Check that the edges join cleanly when copies are placed side by side, and reject any option whose arm directions differ unless the item allows rotation .

Completing a broken figure. To fill a gap in an outline, extend each visible line straight into the gap. The missing piece must match the endpoints and slopes of those lines, and every intersection point must fall inside the gap .

Procedure.

• Identify which attribute changes, and name the rule family.
• Verify the rule on all given figures.
• Eliminate options by one attribute first, then confirm the survivor on the remaining attributes.

Paper folding and cut-out patterns

Paper folding and cut-out patterns is a spatial-reasoning topic in which a sheet of paper is folded, shapes are cut out of the folded stack, and the candidate must predict how the sheet looks after it is unfolded, or decide which folded form or cube net matches a drawing. The governing principle is that folding along a crease is a reflection: each flap lands on the other side of the crease as the mirror image of the part it covers. A cut made through a folded stack therefore reappears on the unfolded sheet at its original place and again at its mirror image across every crease used. A sheet folded k times along independent creases has 2^k layers, so one cut is copied 2^k times, unless the cut touches a crease.

A cut that touches a fold line behaves differently. The two sides of the notch join across the crease, so the unfolded figure is symmetric about that line and the shape straddles the crease instead of forming two separate copies. When the base of a notch lies on an original edge of the sheet, the unfolded result shows a figure symmetric about the midpoint of that edge, so options that place the figure off the midpoint or with the wrong orientation can be eliminated.

A diagonal fold is a reflection across a line at 45^\circ, and it exchanges horizontal and vertical directions. An axis-parallel square stays a square, and a horizontal bar becomes a vertical bar. Because of this, shapes cut from an axis-parallel stack never appear rotated by 45^\circ after a diagonal fold, so a diamond-shaped hole in an option can be rejected at once. Right-angled corners also remain right-angled, which helps check the corners of each candidate.

A reliable procedure is to reverse the folds one at a time: reflect every cut across the last crease, then across the earlier creases, until the sheet is flat. The finished pattern must be symmetric under every reflection and half-turn that the folds generate, so an option that is not symmetric about both diagonals or both midlines cannot be correct. Counting the features and checking one corner is usually faster than drawing the whole pattern.

A cube net is a pattern of six squares joined along edges that folds into a cube. The 1-4-1 type is the most common: a strip of four squares forms four lateral faces around the cube, and one square attached to the top of the strip and one attached to the bottom become the remaining two faces. In the strip, the first and third squares are opposite each other, the second and fourth squares are opposite each other, and the two single squares are opposite each other as well.

Two faces are adjacent when they share a cube edge. In a net, this happens when squares share an edge, or when squares meet at a fold so that they fold onto neighbouring faces. Every face has exactly one opposite face, and the two faces share no edge. Therefore each face is adjacent to four others, and a symbol placed on one face fixes which neighbours can carry which symbols, while the opposite symbol is found by elimination. Students often mistake a square that lies beside a face in the net for its opposite; in a folded strip, the opposite partner is always two places away along the strip.

Symbols and shadings are read from the outside of the cube, and folding keeps each face's marking as seen from outside. A mirror image of the net's marking cannot be produced by folding, because folding never reverses handedness. Only a rotation of the cube is allowed when matching a drawing. A candidate is therefore correct only if some rotation of the cube, not a reflection, shows the same faces as the net.

A shaded region that crosses a shared edge in the net must continue across that same edge on the cube, and a region that lies on two faces must meet along the common cube edge at matching positions. Folding also requires that faces adjacent in the net remain adjacent on the cube, and each shading must keep its orientation relative to the fold lines. These two checks together eliminate most wrong choices.

Each corner of a cube is shared by exactly three faces. A marking placed at a corner of a net face must reappear at the same corner on the other two faces that meet there, in the orientation the fold gives. Checking one vertex this way is often the quickest test, and it also reveals when a cube is a mirror image of the correct one.

When two flaps lie at opposite ends of the strip, they become the top and bottom and are opposite each other. A distinctive marking, such as a corner triangle, fixes the position of its opposite face, and a band on a flap must line up with the lateral face that shares its edge. Starting from the face carrying the most distinctive marking and working outward through its neighbours gives an efficient route to a complete assignment.

In dissection problems, one part of a square is given and the other part is its complement, meaning the cells not covered by the given part. Counting unit cells gives the area of the missing part. The boundary of the empty region, including its concave corners, must match one candidate after a suitable rotation. Candidates whose cell count or outline differ can be rejected without drawing.

• crease — a line along which paper is folded, acting as a mirror line for the flap. 예) Folding a strip along its centre crease copies a notch on one half to the other half.
• mirror image (reflection) — a figure reversed across a line. 예) A letter folded over a vertical line appears reversed on the other layer.
• handedness (chirality) — a property kept by rotation but reversed by reflection. 예) A cube turned around shows the same figure, while a mirror-image cube does not.
• opposite face — the face of a cube that shares no edge with a given face. 예) In a 1-4-1 net, the two single squares become opposite faces.
• complement — the region left when a part is removed from a figure. 예) If a square loses a 3 by 2 block, that block is the complement of the remaining part.

Figure pattern recognition

Figure pattern recognition asks you to find the rule that links a series of figures, tiles or shapes, and then apply that rule to identify the missing item. The rule usually belongs to one of a few families: positional movement, row and column constraints, numerical sequences, transformation (rotation or reflection), and completion of a partly hidden figure. Identify the family first, then test each candidate against every element of the series, not only the nearest one. A rule that fits just two of the given figures is often a coincidence.

Positional movement in a grid. When an element moves by a fixed step from one frame to the next, track its row and column index in every frame. If the movement runs past the edge of the grid, the element usually wraps around to the opposite side, so the row index is taken cyclically. In a four-row grid, row 4 is followed by row 1. Elements that do not move, such as a symbol fixed in one corner, stay unchanged in the answer.

Row and column constraints in tile matrices. In a grid of tiles, a rule often holds along every row and every column. For example, the total count of a property such as filled dots may be the same for each line. A missing count is then found by subtracting the known values from the line total. When two tiles are missing, the answer must satisfy the row and column constraints at the same time, and a pair that satisfies only one of them is rejected.

Consecutive prime numbers. Dot counts sometimes follow the primes 2, 3, 5, 7, 11, 13 and so on. A missing term between two known terms is found by listing the primes that lie between them. Numbers such as 1, 4, 6 and 9 are not prime, so a count of 4 or 6 cannot be the next term of a prime sequence. When a candidate count is not the unique prime in the gap, or repeats an earlier term of a strictly increasing sequence, it is ruled out.

Fibonacci-type sequences. A count sequence such as 1, 1, 2, 3, 5 is generated by adding the two preceding terms. Once this rule is recognized, the next term is the sum of the last two terms, so the missing tile must show that total. Confirm that the rule holds from the first terms before extrapolating, because a sequence that matches only the last few terms is not enough.

Rotation and reflection. A rotation changes only the orientation of a figure; the shape and the relative arrangement of its parts stay the same. A mirror image looks similar but reverses the arrangement, so it is not a rotation even if the pattern seems familiar. When the question specifies a direction and an angle, check whether the candidate is turned clockwise or counterclockwise by exactly that amount, since a turn in the opposite sense is a different answer. For a 45^\circ turn, the sides of a square-based figure are tilted by 45^\circ.

Completing a figure behind a window. When a figure is partly covered by a rectangular window, the correct option must show exactly the parts of each shape that fall inside the window. A cut-off piece must enter through the same edge and sit at the same relative position as in the original figure. Candidates are judged by their overall placement, and an option that shows a complete shape where only a hidden portion should appear is wrong.

• cyclic shift — a movement in which a position that passes the last row or column re-enters at the first one. Example: in a three-row grid, a step from row 3 goes to row 1.
• mirror image — a reflected copy of a figure. Example: the letter F reflected across a vertical line is not the same as the letter F turned by a rotation.
• consecutive primes — prime numbers listed in increasing order with no prime skipped. Example: 5, 7, 11, 13 are consecutive primes, but 5, 7, 13 are not, because 11 is missing.
• Fibonacci-type sequence — a sequence in which each term is the sum of the two preceding terms. Example: 2, 3, 5, 8 continues with 13.

Jigsaw piece assembly

Jigsaw piece assembly asks you to decide which candidate piece, when combined with a given piece or with other pieces, produces a target outline such as a rectangle or a square. Unless the question says otherwise, a candidate may be moved, rotated, or flipped (reflected), and the pieces must meet with neither a gap nor an overlap. The method is to compare edges and boundary profiles, not whole shapes.

Definition and principle. Two pieces fit perfectly when every boundary segment of one piece that lies inside the target is matched by a complementary segment of the other. A protruding part (tab) on one piece must sit in a recess (notch) of exactly the same size and shape on the other. If the target region is T and the two pieces are A and B, then A \cup B = T and A \cap B has zero area. The straight outer edges of the pieces must then continue into the straight sides of the target.

Edge types and how to read them. Typical edges are straight, stepped (zigzag), concave or convex arcs, and notched. A stepped edge is described by the sequence of tooth and notch depths read in one fixed direction along the edge. The missing piece needs the complementary sequence, read in the same direction after the allowed transformation: a tall tooth must face a deep notch, and a shallow notch must receive a short tooth.

Rotation and flipping change the reading. A rotation by 90^\circ or 180^\circ moves an edge to a new side without reversing its order. A flip (mirror reflection) reverses the order of the profile along the edge, so a profile that reads 2-1-3 from one end reads 3-1-2 from the other end. Checking a candidate therefore means testing its complement in every allowed orientation, not only in the orientation in which it is first drawn.

Confusable cases. Two failures look alike but have different causes. A gap remains when the candidate's edge is too short or too shallow to fill a notch, while an overlap occurs when its tab is too large and projects into the given piece. Both are rejected. An edge that is identical to the given edge is also wrong, because two identical profiles would collide rather than interlock.

Quick checks before deciding.

• Count the right-angle corners of the target. A rectangle or square needs four, and each corner must be supplied by a piece that carries a convex corner there.
• Count cells (area check). If the pieces are built from unit squares, the total cell count must equal the target cell count. When every component is a fixed group of 4 cells, the target area must be a multiple of 4, so a figure whose area is not a multiple of 4 cannot be covered. For example, a 3\times3 block with 9 cells cannot be covered by 2\times2 patterns alone.

Holes formed by arcs. A round hole in an assembled figure is closed only when concave arcs of equal radius from neighbouring pieces meet and close into one circle, while a convex bump on one piece must face a bite on another. Count closed cavities, not partial arcs: one complete circle gives one hole. Also confirm that the outer boundary consists of straight sides meeting at right angles.

Hidden part of a cut figure. When a figure is cut into two parts, the second part is exactly the region of the whole that the first part does not cover, so the two areas add up to the area of the whole. Compare the cells along the cut line: the boundary of the given part, traced across the cut, must continue into the boundary of the other part.

Silhouettes and missing fragments. For a silhouette, the outline of the target equals the union of all pieces, each used exactly once, so every concave notch of the outline must be filled by a convex corner of some piece. For a missing fragment, extend the visible lines and curves across the gap; the missing piece must have edges whose continuation joins the visible outline without a kink or a break.

Key expressions.

• tab — a protruding part of a piece that fits into a recess. 예) The tab on the left piece slides into the notch on the right piece, and the two outlines join along a straight line.
• notch (recess) — a cut-in part of an edge that receives a tab. 예) A deep notch needs a tab of the same depth, so a shallow tab leaves a gap.
• complement — the shape or edge that exactly fills what another piece leaves empty. 예) The complement of a 2\times3 hole is a 2\times3 block.
• interlock — two edges mesh so that no gap remains. 예) Stepped edges interlock only when their step depths are reversed.
• reflect / flip — produce the mirror image, reversing the order of an edge profile. 예) A flipped piece reads its profile in the opposite direction.
• bounding box — the smallest rectangle that contains a figure; aligning it with the target makes edge comparison faster. 예) Align the bounding box of the candidate with the target's corner before testing the edges.

3D block projection view

A view of a three-dimensional block is the two-dimensional picture of the solid as seen along one fixed direction, where all lines of sight are parallel to that direction. In an orthographic view the picture plane is perpendicular to the viewing arrow, so a face parallel to the plane appears in its true shape, while an inclined face appears foreshortened, or as a line when it is seen edge-on.

To choose the correct view, first fix the outline. The overall width and height of the view are the extents of the solid in the two directions perpendicular to the arrow, and each block contributes its own profile at the level where it sits, so a trapezoidal block standing on a rectangular block appears as a trapezoid above a rectangle. Then mark the edges that separate faces, and remember that an edge hidden behind material is not drawn as a visible line. A line crossing the base in a view must therefore correspond to a real visible edge of the solid, and if no such edge exists the option is wrong.

Looking from the opposite side reverses the left and right of a view, so a sloping face that appears on the left from one side appears on the right from the other. Distractors that are mirror images of the correct figure are common, and they are checked by asking on which side of the arrow the sloping face lies.

• orthographic view — a projection along one fixed direction with parallel lines of sight. Example: a block with one sloping face shows that face as a single slanted edge in the front view.
• foreshortening — the shortening of a face that is inclined to the picture plane. Example: a tilted square appears as a narrower rectangle.
• hidden edge — an edge behind the solid that is not drawn as a solid line.
• mirror-image view — the same solid seen from the opposite side, with left and right interchanged.

Shadows cast by a parallel beam of light are projections of the same kind, since every point of the object is carried along the light direction onto the screen. For a convex solid the shadow is the convex hull of the projected vertices, so it is always convex and can never contain a re-entrant notch; a figure with a notch therefore rules out a convex object.

The shadow of a circular cylinder has a smooth boundary. Light along the axis gives a circle, oblique light gives a rounded shape bounded by two elliptical arcs joined by two parallel tangent lines, and light lying in the plane of one end face gives a rectangle. A figure with sharp corners at non-right angles cannot be the shadow of a cylinder.

A top view of a hill is drawn as contour curves, one for each height, where each closed curve encloses the points at or above that height. The innermost curve marks the summit, and the outer curves enlarge in turn. Close spacing of contours shows a steep side, while wide spacing shows a gentle side. Because the centre of each horizontal section moves as the height changes, the innermost contour lies off the centre of the outer contour whenever the hill is not symmetrical.

When an object is built from identical cubes, one view fixes the number of cubes in each column, which narrows the possible arrangements before the other view is drawn. Count the visible cube faces in each column and check that no cube is missing from a stack.

A smaller cube whose face is parallel to the picture plane appears as a smaller square, and its position is found by matching the outline of the assembly rather than its interior.

A three-dimensional structure of unit blocks is identified from two orthographic views. The top view gives the footprint of occupied cells, and the front view gives the height profile of each column. A candidate is accepted only if both projections match exactly, and a solid that agrees with one view but not the other is eliminated.

Neighbour cross count in grid

A neighbour cross count grid places numbers in some cells, and each number states how many crosses lie among the immediate neighbours of that cell. The neighbourhood is the eight cells that share an edge or a corner with the cell: left, right, top, bottom and the four diagonals. The cell itself is never counted, and positions outside the grid do not exist, so an edge cell has fewer neighbours than an interior cell.

The usual method is to name each unknown cell with an indicator, where 1 means a cross and 0 means no cross, and to write one equation per numbered cell. The known crosses among the neighbours are subtracted from the number, leaving a sum of unknown indicators equal to a small integer. Two numbered cells in adjacent rows share most of their neighbours, so subtracting one equation from the other isolates a single unknown, and this is how a cell is forced to be empty.

When a question asks for the maximum number of crosses in a column, solve the constraints first and then maximize the sum. An equation of the form a+b=1 allows exactly two solutions, so the pair contributes at most one cross to the total. Example: if one number gives a+b=1 and another gives b+d=1, then a+b+d is at most 2, reached with a=d=1 and b=0. Before accepting a configuration, check every numbered cell, not only the ones used to derive the equations.

• cross indicator — a variable that equals 1 for a cross and 0 for a blank cell.
• forced blank cell — a cell that the equations force to be empty, usually because a small number already uses up all its allowed crosses.
• constraint isolation — subtracting two neighbour equations to find the value of one unknown.

The maximum depends on which constraints are binding. Constraints that overlap leave some cells free, and the correct count is the largest total that satisfies every equation at once.

A numbered cell in the column next to the empty column often decides its neighbours. When the number equals the count of known crosses plus one, exactly one neighbour in the empty column must be a cross, and when it equals the known count, all those neighbours must be blank. Combining such conditions from several rows limits which cells may be crosses at the same time.

When several numbers constrain the same cells, the constraints must be satisfied together, so the feasible set is the intersection of their solutions. Testing each candidate pattern against all the numbers removes any solution that violates one of them.

Some puzzles count the non-cross neighbours instead of the crosses. A number then counts the blank cells among its neighbours, and a numbered cell is itself a non-cross because it is not a cross. When a row contains no crosses, every cell in that row is a non-cross, and each value is found by counting its non-cross neighbours within the grid directly, not by solving equations. The total of the row is the sum of these counts.

Lines of symmetry in unit-square figure

A line of symmetry is a line that divides a figure into two parts such that each part is the mirror image of the other across that line. For a pattern built from unit squares, the test is to imagine folding the figure along the line: every black square must land exactly on another black square. A square whose centre lies on the axis is its own mirror image and therefore counts once, whereas every square off the axis pairs with a square at the same distance on the opposite side, so off-axis squares are added in pairs.

When a pattern must have two lines of symmetry, the requirements interact. If a figure is symmetric about line L1 and about line L2, it is also symmetric about the composition of the two reflections, so the set of black squares must be closed under every reflection that the two lines generate. Two axes that meet at 45^\circ, such as a diagonal and a midline of a square grid, generate the full group of eight symmetries of the square, including rotations by 90^\circ about the centre.

This closure produces orbits. An orbit is the set of cells to which one cell can be carried by the symmetries of the figure. In a 4\times4 grid the sixteen cells split into three orbits: the four corner cells, the four central cells, and the eight edge-middle cells. A symmetric colouring must be a union of whole orbits, so leaving even one cell of an orbit white breaks the symmetry.

The usual exam question asks for the minimum number of squares to add. The method is to find the orbit that contains the existing black squares. If all the given black squares lie in one orbit, that whole orbit must be black, and the number of squares to add is the orbit size minus the number already black. For example, if an orbit of eight cells already contains three black cells, then 8 - 3 = 5 more are needed. Because every orbit in this grid has size four or eight, a symmetric pattern always has a black count divisible by four, so totals such as six, seven or nine can never be symmetric.

Key terms:

• mirror image — a figure that matches another when reflected across a line. Ex) In a 3\times3 grid with a vertical axis, a black cell in the left column needs a black partner in the right column in the same row.
• axis of symmetry — the line about which the figure is reflected onto itself. Ex) The midline of an even-width grid lies between two columns, so no cell sits on it.
• orbit — the group of cells equivalent under all symmetries of the figure. Ex) The four corners of a 4\times4 grid form one orbit of size four.
• closure — the property that applying any symmetry to a black set keeps it inside the set. Ex) Reflecting a black set in two axes and then combining the results must still give a black set.

Tiling a plane with a shape

A shape tiles (tessellates) the plane if identical copies of it can cover the whole plane, extending to infinity in all directions, with no gaps and no overlaps. For a polygon whose copies meet edge to edge, the basic test is that the interior angles of the copies meeting at each vertex must add up to exactly 360^\circ, so that the copies fit around every vertex without leaving a hole.

A regular n-gon has interior angle \frac{(n-2)\cdot180^\circ}{n}. A regular polygon tiles the plane alone only if 360^\circ divided by its interior angle is a whole number. This holds for the equilateral triangle (60^\circ), the square (90^\circ) and the regular hexagon (120^\circ). It fails for the regular pentagon, whose angle is 108^\circ, and for the regular octagon, whose angle is 135^\circ, because neither 360/108 nor 360/135 is an integer. A circle never tiles the plane, since its curved boundary always leaves gaps between neighbouring copies. (Note that some non-regular pentagons do tile the plane, so a pentagon is excluded only in its regular form.)

A rhombus, being a parallelogram, tiles the plane. Its opposite sides are parallel and equal, so copies translated along two independent directions fill the plane without gaps or overlaps. At each vertex the four angles of a rhombus, \theta, 180^\circ-\theta, \theta and 180^\circ-\theta, sum to 360^\circ, which is the condition the vertex must satisfy. More generally, every triangle and every quadrilateral tiles the plane, including copies rotated by half-turns about edge midpoints.

Tiling a region with pieces is a different counting problem. If each piece covers a fixed number of unit tiles, then any region covered exactly by such pieces, with no overlaps, must contain a total number of unit tiles that is a multiple of the piece size. A region whose count is not a multiple of the piece size can be excluded immediately, without drawing any arrangement.

Key terms:

• tile / tessellation — a shape whose copies cover a plane or region without gaps or overlaps. Ex) A square is a tile, and a 2\times2 block of squares is a larger tile of area four.
• interior angle — the angle inside a polygon at one of its vertices. Ex) The interior angle of a square is 90^\circ, so four squares meet at a vertex.
• translation — sliding a copy without rotating it. Ex) Translated copies of a rhombus form a lattice that covers the plane.
• gap / overlap — empty space or doubly covered space between copies, both forbidden in a tiling. Ex) Two circles placed in a grid leave curved gaps between them.

Mirror image of text about an axis

A mirror image about an axis is a reflection. Every point moves to the opposite side of the axis at the same distance, so the size and shape of each glyph are kept while its orientation is flipped. For text, the axis decides what happens to the string. A reflection about the X-axis, the horizontal axis of the figure, flips each glyph top to bottom but keeps the left-to-right order of the characters. A reflection about the Y-axis does the opposite: it reverses the order of the characters and flips each glyph left to right. A rotation by 180 degrees combines both changes.