Concept 3 of 7
Symbol Equations: When Shapes Stand for Numbers
Today's big idea
Solve the equation with a single kind of shape first, then substitute.
Once one shape's value is known, swap it into the next equation to unlock the rest.
How it works
👀 What's going on?
Some puzzles hide numbers behind shapes. Each shape is worth a fixed amount, and you are handed just enough equations to pin every one of them down.
💡 Here's the trick
Look at the pair above. The first equation holds only stars, so it can be solved on its own: two stars make 8, so one star is 4. Drop that 4 into the second equation and the triangle falls out as 6. Always begin with the equation holding a single kind of shape.
🪄 How to do it
- 1.Find the equation with only one kind of shape, as the top one in the picture above.
- 2.Solve it, then substitute that value into the next equation.
- 3.Repeat until every shape has a value.
✍️ Try it
Four circles make 24, and one circle plus one square makes 10. What is each shape worth?
Four circles make 24, so a single circle is 24 divided by 4 = 6. Putting that into the second equation gives 6 + square = 10, so the square is 4. Check it: 6 + 4 really does make 10.
⚠️ Watch out
You might be thinking you can start with whichever equation looks easiest. Start with the one holding a single kind of shape. An equation with two unknowns cannot be solved alone, however friendly its numbers look.
🌟 Why this matters
This is algebra with the frightening letters swapped for shapes. Working out something unknown from what you already know is a move you will use for years.
Exam tip: Substitute your answer back into the original equation, because it takes seconds and catches an arithmetic slip.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“If ★ + ★ + ★ = 15 and ★ + △ = 8, find △.”
- 1
Solve simpler one
3★ = 15 → ★ = 5.
- 2
Substitute
5 + △ = 8
- 3
Solve
△ = 3
- 4
Answer ✓
△ = 3
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
Solve the equation with only ★ first, then put its value into the other one.
SYMBOL EQUATION · ROUND 1 OF 3
★ + ★ + ★ + ★ = 20
★ + △ = 9
Solve the first equation: ★ = ?
Quick check
✅ VERIFICATION
Three squares make 21, and one square plus one circle makes 12. What is the circle worth?
⚡ Finish the activity above first to answer.
Keep going!
Up next: Substitution Chain
Answer normally → look up the rename rule → final answer.