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Free guide · Spatial reasoning · 5 min read

How to solve figure assembly puzzles

Which piece completes the square?

Match the hole's outline feature by feature — cell counts first, then notch positions — and watch for forbidden flips.

Updated

A square with a bite taken out, and candidate pieces — but read the question, because the same picture carries six of them: slide one piece in with no turning; fill the hole with turning allowed; fit a piece after exactly one quarter-turn; find the piece that could NOT be cut from the hole; pick the SET of three pieces that tiles it; or name the piece left over once the others have. The failure mode is gestalt guessing ('that one looks right'). The method is feature accounting: cells, notches, orientation.

Count first — cells are the accounting unit

Count the hole's cells (squares). On one-piece questions, any piece with a different cell count is dead before shape is even considered. On three-piece questions, area accounting does the heavy lifting: the three pieces' cells must SUM to the hole's cells, which usually kills a wrong set outright — and on leftover questions, the leftover piece's count is the total minus the hole.

Orientation discipline

Read the stem: if pieces may NOT be turned, the piece must match the hole exactly as drawn — rotated copies of the right shape are traps. If turning IS allowed, mirrors become the traps: a flipped piece fits only if you turn it over, which the rules forbid. If the stem prescribes exactly ONE quarter-turn clockwise, the direction and the amount both matter — the same piece turned the other way is on offer.

Intruders and tilings

'Which piece cannot be cut from the hole?' is a containment hunt: try each piece in every position and turn, and eliminate the ones that fit somewhere — the answer is the one that fits NOWHERE. For three-piece tilings, place the most awkward piece first (the one with a notch or a long arm), anchor it against the hole's most constrained corner, and let the remaining space dictate the rest.

The strategy

  1. Read the question: one piece or three, turning allowed or not, fit or intruder or leftover.
  2. Count the hole's cells; discard options (or sets) whose cell arithmetic fails.
  3. Describe the hole row by row ('two cells, then a three-cell row flush left').
  4. Find the option matching that description in the allowed orientation.
  5. Confirm the notch positions — near-miss pieces differ by exactly one cell.

Common traps

  • Rotated versions when turning is forbidden; mirrored versions when it is allowed.
  • The prescribed quarter-turn made in the wrong direction.
  • One-cell-off shapes that pass a lazy glance.
  • On set questions, checking shapes before checking that the areas add up.

Speed · ~35s/item: Cell counting plus one row-by-row description: 15–40 seconds for one-piece questions; give tilings and intruder hunts up to a minute.

Worked example

Made by the same generator and answer checker as the practice questions, at middling difficulty, but never served in a test. Commit to an option before you reveal — the pick is yours alone and is never recorded.

Turn each piece ONE quarter turn clockwise (90° ↻). Which piece then fits the hole exactly?

Reveal the answer

The answer is C.

Show why — the worked reasoning
Turn each option one quarter turn CLOCKWISE in your head, then compare it with the hole. The reliable trick is to track one landmark: pick the piece's longest straight edge, note which way it points, and remember that a quarter turn clockwise sends up→right, right→down, down→left, left→up. Three of the options are the right shape turned by the wrong amount — one of them already matches the hole as it is drawn, which is the trap for anyone who forgets to turn it at all — and one option is the wrong shape altogether.

Try it yourself: 4 practice questions

Pick an answer, then reveal it. Nothing is recorded, there is no clock, and none of these questions is ever used in a test.

Question 1 · Easy

Turn each piece ONE quarter turn clockwise (90° ↻). Which piece then fits the hole exactly?

Reveal the answer

The answer is E.

Show why — the worked reasoning
Turn each option one quarter turn CLOCKWISE in your head, then compare it with the hole. The reliable trick is to track one landmark: pick the piece's longest straight edge, note which way it points, and remember that a quarter turn clockwise sends up→right, right→down, down→left, left→up. Three of the options are the right shape turned by the wrong amount — one of them already matches the hole as it is drawn, which is the trap for anyone who forgets to turn it at all — and one option is the wrong shape altogether.
Question 2 · Easy

Turn each piece ONE quarter turn clockwise (90° ↻). Which piece then fits the hole exactly?

Reveal the answer

The answer is D.

Show why — the worked reasoning
Turn each option one quarter turn CLOCKWISE in your head, then compare it with the hole. The reliable trick is to track one landmark: pick the piece's longest straight edge, note which way it points, and remember that a quarter turn clockwise sends up→right, right→down, down→left, left→up. Three of the options are the right shape turned by the wrong amount — one of them already matches the hole as it is drawn, which is the trap for anyone who forgets to turn it at all — and one option is the wrong shape altogether.
Question 3 · Easy

Turn each piece ONE quarter turn clockwise (90° ↻). Which piece then fits the hole exactly?

Reveal the answer

The answer is C.

Show why — the worked reasoning
Turn each option one quarter turn CLOCKWISE in your head, then compare it with the hole. The reliable trick is to track one landmark: pick the piece's longest straight edge, note which way it points, and remember that a quarter turn clockwise sends up→right, right→down, down→left, left→up. Three of the options are the right shape turned by the wrong amount — one of them already matches the hole as it is drawn, which is the trap for anyone who forgets to turn it at all — and one option is the wrong shape altogether.
Question 4 · Easy

Which piece slides straight into the hole, without being turned or flipped?

Reveal the answer

The answer is E.

Show why — the worked reasoning
The piece must slide straight in without turning. Compare the hole row by row: count how many cells it spans in each row and where the notches sit. Rotated and mirrored copies of the right shape are the classic traps here — they match the cell count but not the orientation.

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Will figure assembly questions be on the free test?

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