Problem
Find a number meeting given conditions
Circle ●, square ■ and triangle ▲ each stand for a different number from 1 to 9. Three equations are given. Find the number behind each shape.
Reasoning · Grade 2-2 Rules and Defined Operations
Your answer
Watch out Reading a value from one equation leaves many candidates — two equations together pin a shape down.
How to solve
Strategy Identify Subproblems — Instead of guessing all three shapes at once, compare the two addition/subtraction equations to pin down one shape first, then use that to break the remaining work into easy single-shape steps and finish with the multiplication equation.
1STEP 1
Find the square by comparing the first two equations
Stacking two equations cancels a shape and gives ■ = 2.
(● + ■ - ▲) - (● - ■ - ▲) = 4 - 0 → 2 ■ = 4 → ■ = 2
Lining the two equations up so the matching parts cancel isolates the square all by itself.
1.OA.D.8Identify SubproblemsLining the first two equations up side by side lets the matching parts cancel and leaves the square standing alone.
Why?
Removing the same amount from both sides of a true equation keeps it true, so a shared part can simply be struck out.
🧱Do the same to both sides and equality holdsA balanced scale stays level when you take the same amount off both pans or add the same amount to both.
Why?
Each side is its pieces added together, so what remains after the shared pieces go is exactly the pieces that differ.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
2STEP 2
Relate the circle and the triangle
Substituting shows ● = ▲ + 2.
● - 2 - ▲ = 0 → ● = ▲ + 2
Knowing the square lets us rewrite one equation as a simple 'circle = triangle + 2' relationship.
1.OA.D.8Identify Subproblems3STEP 3
Use the product to find the triangle and circle
The pair whose product is 24 gives ▲ = 4, ● = 6.
● × ▲ × 2 = 48 → ● × ▲ = 24, ▲=4, ●=6: 6 × 4 = 24
We only need two numbers 2 apart whose product is 24, and 6 and 4 are the only single-digit pair that fits.
3.OA.A.4Guess And CheckAnswer
● = 6, ■ = 2, ▲ = 4
Check all three: 6 + 2 − 4 = 4 ✓; 6 − 2 − 4 = 0 ✓; 6 × 4 × 2 = 48 ✓. The three values 6, 2, 4 are all different single-digit numbers, as required.
Takeaway
Stacking two equations so parts cancel finds one shape instantly — then the rest falls out with a quick times-table check you already know!
- Find the square by comparing the first two equations
- Relate the circle and the triangle
- Use the product to find the triangle and circle