How to solve shape algebra puzzles
Shape algebra: start where one unknown lives
A grid of shapes with row and column totals is a system of equations. Solve the line with the fewest distinct shapes, substitute, repeat.
Updated
Three circles total 15, so a circle is 5 — shape-value puzzles are simultaneous equations wearing a costume. They reward order of attack: usually one row or column yields a value quickly, and each solved value simplifies another line. Reading order is the enemy; dependency order is the method. But be warned: some items are built so that NO single value can be pinned down — only a combination is — and the question asks for exactly that combination.
Find the entry line
Scan every row and column for the one made of a single repeated shape ('3 squares = 12') or one solved shape plus one unknown. That line is the entry point. No such line? Look for two lines differing by exactly one shape — subtracting them isolates that shape's value.
Substitute and cascade
Write each solved value into every line containing that shape. Each substitution usually reduces some other line to one unknown. The whole grid falls in a chain — circle → square → triangle — never all at once.
Answer what CAN be known
Some questions never ask for a single value at all. 'What is circle + triangle?' may be answerable even when neither shape alone is — the equations fix only the combination, and grinding for individual values wastes the clock. Others ask which symbol is LARGEST — solve the little system, then rank the values — or which comparison MUST be true, which is pure ordering: chain the given comparisons and see what follows from them alone. And when one equation merely repeats what the others already say, exactly one symbol may still be pinned; the question asks which. In every case, the stem tells you what is knowable — trust it and solve for that.
The strategy
- Scan all totals for the single-shape or one-unknown line. Solve it.
- Propagate the value everywhere before hunting the next line.
- Answer exactly what the query asks — a sum, a single shape, a missing total.
- Verify with one line you never used: if it balances, the solution is certain.
Common traps
- Solving top-to-bottom because it's first, and getting stuck on a three-unknown row.
- Query asks for ○+△ and the options include the value of ○ alone — answer the question asked; on some items the combination is the ONLY thing the equations pin down.
- Sign slips when a line mixes addition and subtraction.
- Skipping the free verification line and locking in an arithmetic slip.
Speed · ~50s/item: Entry line + cascade solves most grids in under a minute. The verification line costs five seconds and refunds whole points.
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.
Every copy of a symbol has the same value. What is the value of the asked symbol?
Reveal the answer
The answer is C — 4.
Show why — the worked reasoning
Every copy of a symbol has the same value. What is the value of the asked symbol?
Reveal the answer
The answer is C — 8.
Show why — the worked reasoning
Every copy of a symbol has the same value. What is the value of the asked symbol?
Reveal the answer
The answer is D — 8.
Show why — the worked reasoning
Every copy of a symbol has the same value. What is the value of the asked symbol?
Reveal the answer
The answer is A — 8.
Show why — the worked reasoning
Try questions like this
The free test has 20 questions across abstract, numerical, verbal and spatial reasoning. It takes 12 minutes, needs no sign-up, and explains every answer.
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Questions people ask
Will shape algebra questions be on the free test?
Maybe. The free test gives numerical reasoning 5 of its 20 questions, spread across 13 different question types, so any one test may or may not include this kind.