Shape equation puzzles
Use balanced equations to find the value of each shape.
Example
Example
Example
An example to explore. No score or practice round is recorded.
6 questions · Explanation after each answer
6 questions · No time limit
- Answered
- 0 / 6
- Active time
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Try six questions.
The same shape always has the same value. Use every equation to find the requested value.
Possible changes
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Explore the method
Find one shape's value, then use it in the next balance.
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How this data is usedHelp
The same shape has the same value throughout a question, and both sides of each equation have equal value. Start with an equation that reveals one shape, then substitute that value into the others. Multiplication means repeated copies of a shape. Use every equation and look for a whole-number solution.
Select one of five answers, then choose Check. You can change your selection before checking. Read the explanation and continue when ready. Keyboard: Tab to the answers and use the arrow keys to change your choice.
Solving time excludes setup, explanations and background tabs. The optional challenge gives you two minutes of solving time across six questions. When time runs out, unanswered questions count as incorrect. Timed and untimed results stay separate.
The browser loads a complete six-question round from the server when you start or restart. Completed rounds and timed-out challenges are saved when browser storage is available; ending early is not saved. Your individual answers are not uploaded.
This is a practice activity, not a standardized or diagnostic assessment. Results describe only this activity.
Keyboard: press an answer's letter or number to choose it, Enter to check, and Enter again to continue.
About this practice
Use each set of balanced shape equations to find the requested value. The same symbol keeps the same value, and every answer includes a worked explanation.
Each shape stands for a number. Within one question, every copy of the same shape has the same value, and both sides of every equation must be equal. Use the equations to find the value asked for, which may be one shape or a total involving several shapes. Choose from 5 answers and check to see the explanation.
Find one value, then substitute it
- Two identical circles total 8, so each circle is 8 ÷ 2 = 4.
- Replace the circle in the second equation with 4. The triangle is 11 − 4 = 7.
- Check both equations: 4 + 4 = 8, and 4 + 7 = 11. Answer: 7.
Do not split 11 equally between the circle and triangle. Different shapes can have different values. A new question can assign new values, so start from its equations each time. A number multiplying a shape counts copies: 2 × circle means two circles.
When no shape is given directly
Compare whole equations. If one contains two circles plus a square, and another contains one circle plus the same square, subtracting the second equation from the first removes the square and one circle. The difference between their totals is the remaining circle's value. Subtract both sides, then substitute and check every original equation.
What each difficulty changes
- Easy: divide equal groups, substitute one known shape, or use a sum and a difference together to find two values.
- Standard: follow a chain of substitutions, subtract equations with a shared term, or use one shape as a multiple of another. The requested answer can combine shape values.
- Challenge: combine three pairwise sums, eliminate values from equations with different multipliers, or follow fixed differences between shapes. For three pairwise sums, adding all three totals counts every shape twice, so halve the sum to get the total of the three shapes.
The current questions use positive whole-number shape values. The harder levels change the relationships you must combine, not just the size of the numbers. After finding the individual values, reread the question before choosing: it may ask for a total such as two copies of one shape plus another.
What your result means
A round has six questions. Choose No time limit or the optional 2-minute challenge. The timer counts solving time across the whole round, excluding setup, explanations and background tabs. If it expires, unanswered questions count as incorrect. Speed adds no points.
The score counts correct answers out of six. Completed rounds and timed-out challenges can be saved when browser storage is available; ending early is not saved. Compare the same difficulty and timing mode, then use the explanations to check whether an error came from finding a value, substituting it or answering the requested total. Six questions describe this round, not your overall reasoning ability.
To practice substitution step by step, try Keep both sides equal.
For a broader mix of practices, explore the reasoning practice test. Sign in free to start a test.
Report a problem
On by default unless you turn it off. Saved choices are kept. Shares starts, completions, replays and error categories. For ChatGPT visits, starts and completions may include the first QuickenBrain page and country. No answers, drawings or names.
How this data is used