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Lines of symmetry

Does the pattern fold onto itself along an upright line, a flat one, both or neither? Later: how many ways does it turn onto itself?

  • Spatial
  • Timed
  • Endless levels

How it plays

A pattern of dark squares. Imagine folding it along the upright line through its middle: if every dark square lands on another, that line is a line of symmetry. Then do the same with the flat line. Say which work — upright, flat, both or neither.

Later questions ask about turning instead: spin the pattern about its centre and count how many positions in a full turn look exactly the same — 1, 2 or 4.

Levels grow the board without end, then bring near misses — a symmetric pattern with one square changed — then shorten the time. The working names the square that breaks a symmetry.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This practises judging reflection and rotation symmetry in patterns like these, faster and more carefully. It claims nothing about spatial ability in general.

Learn

How to get better at lines of symmetry: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Find the pattern’s own middle: the box from its topmost to its bottommost dark square and its leftmost to its rightmost. The lines and the centre are measured on that box, not on the board.
  2. Test the upright line: is every row of the box the same read from either end? One row that is not settles it — no upright line.
  3. Test the flat line: does the top row of the box match the bottom row, the second the second-last, and so on in towards the middle?
  4. For a turning question, test the half turn first, then the quarter turn, and answer with the highest that holds: 4, 2 or 1.

Techniques and tricks

  • Rows as palindromes

    An upright line of symmetry means every row reads the same forwards and backwards. Say each row as dark and light — “dark, light, light, dark” — and listen for the echo.

    e.g. ■□□■ passes; ■■□■ fails

  • Pair the rows from outside in

    A flat line of symmetry means the first and last rows are identical, then the second and second-last. Compare the outermost pair first — that is where a stray square is easiest to see.

  • Half turn: last row backwards

    For a half turn, the top row must equal the bottom row read backwards, the second row the second-last read backwards, and so on.

    e.g. Top ■■□, bottom □■■: that pair passes the half turn

  • Both lines give a half turn free

    Flipping in the upright line and then in the flat line is the same as a half turn. So a pattern with both lines always looks the same after a half turn. A half turn can still hold with no lines at all, like an S.

  • Quarter turns need a square

    A quarter turn swaps width and height, so it can only fit if the pattern’s box is square. And anything that survives a quarter turn survives a half turn too — check the half turn first.

  • One break is enough

    A symmetry fails if a single dark square lands on a light one. Do not weigh how symmetric it looks overall; look for the one square that breaks it.

As the levels climb

  • The board grows a row and a column each round of levels, without end. On a big board, find the pattern’s box first and work only inside it.
  • Near misses arrive: a symmetric pattern with one square, or one pair, changed. From here “nearly symmetric” is the common case, so check every row pair, not a few.
  • Then turning questions join the line questions — unless you asked only for one kind. Read the question each time: lines and turns have different answers for the same pattern.
  • The clock closes in three steps, to 80%, 65% and 55%. Under the tight clock, test the outermost rows and columns first, since they are quickest to compare, and stop at the first break.

Traps to avoid

  • Judging against the board’s middle instead of the pattern’s. A pattern that sits off-centre on the board can still be perfectly symmetric about its own middle.
  • A near miss: one square out in an otherwise perfect pattern. The overall look says “symmetric”; only a row-by-row check finds the square, and the working names it.
  • Answering “Both” for a pattern that only turns onto itself. A half turn is not a line of symmetry — test each line separately.
  • Answering 4 when the box is not square, or 2 for a pattern that also survives a quarter turn. Give the highest order that holds.

How to practise

  • Tick only “Lines of symmetry” under Questions until the row checks are quick, then only “Turning symmetry” to drill the half and quarter turns, then both.
  • Say the rows aloud as dark and light for a while, even when the answer looks obvious. The near misses are built to look obvious.
  • When you miss, read the working: it gives the dark square that lands on an empty one, by row and column. Find it on the pattern before going on.
  • Set Time pressure to “Relaxed — 1.5× time” while the row checks are new, and back to Normal once you stop missing near misses.

Levels

Each question has a clock that comes from the work it asks for. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

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