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Picture Equation Balance Puzzles

Picture equations are balance puzzles: if both sides of the scale match, both sides have the same total value. Inside one puzzle, identical shapes always stand for identical values.

This page is designed as 4th-grade enrichment / early pre-algebra. Kids can practice removing known unit cubes, spotting equal groups, and combining balances without needing formal algebra notation first.

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Five picture-equation examples

Start with the fixed examples below, then try the seeded generator for more balance puzzles.

Example 1: One unknown — remove known units

Easy
2 circles + 3 unit cubes = 13 unit cubes
What is the value of 1 circle?
Expression: circle
Reveal worked steps

Worked steps

  1. Take away 3 unit cubes from both sides. That leaves 2 circles = 10.
  2. Split 10 equally between the 2 matching circles. Each circle = 5.
  3. Check: 2 × 5 + 3 = 13.
Answer key
Values: circle = 5
What is the value of 1 circle? 5

Example 2: Known substitution — two shapes

Medium
3 triangles + 2 squares = 18 unit cubes
1 square = 3 unit cubes
What is the value of 1 triangle?
Expression: triangle
Reveal worked steps

Worked steps

  1. Use the second balance first: 1 square = 3, so 2 squares = 6.
  2. Replace the 2 squares in the first balance with 6. Now 3 triangles + 6 = 18.
  3. Subtract 6 from 18. That leaves 3 triangles = 12.
  4. Split 12 into 3 equal groups. Each triangle = 4.
  5. Check: 3 × 4 + 2 × 3 = 18.
Answer key
Values: triangle = 4, square = 3
What is the value of 1 triangle? 4

Example 3: Two equations, two unknowns

Medium
2 stars + 1 moon = 13 unit cubes
1 star + 2 moons = 14 unit cubes
What is star + moon?
Expression: star + moon
Reveal worked steps

Worked steps

  1. Double the second equation: 2 stars + 4 moons = 28.
  2. Subtract the first equation, 2 stars + 1 moon = 13. That leaves 3 moons = 15, so 1 moon = 5.
  3. Put moon = 5 back into 2 stars + 1 moon = 13. Then 2 stars = 8, so 1 star = 4.
  4. Now add them: star + moon = 4 + 5 = 9.
  5. Check both balances: 2 × 4 + 5 = 13 and 4 + 2 × 5 = 14.
Answer key
Values: star = 4, moon = 5
What is star + moon? 9

Example 4: Equivalent groups / ratio challenge

Medium
2 diamonds = 3 circles
1 diamond + 1 circle = 20 unit cubes
What is diamond - circle?
Expression: diamond - circle
Reveal worked steps

Worked steps

  1. From 2 diamonds = 3 circles, think in equal groups: each diamond can be 3 equal groups, and each circle can be 2 equal groups. Then 2 diamonds = 6 groups and 3 circles = 6 groups.
  2. That means diamond + circle = 3 groups + 2 groups = 5 groups.
  3. The second balance says diamond + circle = 20, so 5 groups = 20. Each group = 4.
  4. A diamond is 3 groups, so diamond = 12. A circle is 2 groups, so circle = 8.
  5. Now subtract: diamond - circle = 12 - 8 = 4.
  6. Check: 2 × 12 = 24 and 3 × 8 = 24, and 12 + 8 = 20.
Answer key
Values: diamond = 12, circle = 8
What is diamond - circle? 4

Example 5: Three-picture challenge

Challenge
2 hexagons + 1 triangle = 16 unit cubes
1 triangle + 2 circles = 14 unit cubes
1 hexagon + 1 circle = 11 unit cubes
What is hexagon + triangle + circle?
Expression: hexagon + triangle + circle
Reveal worked steps

Worked steps

  1. Add the first two equations. That gives 2 hexagons + 2 triangles + 2 circles = 30.
  2. Double the third equation: 2 hexagons + 2 circles = 22.
  3. Subtract to find 2 triangles = 8, so 1 triangle = 4.
  4. Put triangle = 4 into triangle + 2 circles = 14. Then 2 circles = 10, so 1 circle = 5.
  5. Use hexagon + circle = 11. Since circle = 5, hexagon = 6.
  6. Now add them: 6 + 4 + 5 = 15.
  7. Check all three balances: 2 × 6 + 4 = 16, 4 + 2 × 5 = 14, and 6 + 5 = 11.
Answer key
Values: hexagon = 6, triangle = 4, circle = 5
What is hexagon + triangle + circle? 15

Try the generator

Pick Easy, Medium, or Challenge. Use a seed when you want the exact same puzzle again for class review, homework, or testing.

Make a fresh puzzle

Seed: easy-starter
Easy
4 stars + 12 unit cubes = 44 unit cubes
What is the value of 1 star?
Expression: star

FAQ