Solve the puzzle and discuss your strategy with your friends. Can you determine the value represented by each shape?
Step-by-step solution
Idea: Each row or column with a total is an equation. Look for two equations that share a block of shapes: the first row gives T + R = 14, and the last column contains T + R as well, so the circle can be found at once. Then the rest follow one by one.
- Let triangle = T, rectangle = R, circle = C, hexagon = H. Row 1: 2T + 2R = 28, so T + R = 14.½ mark
- Last column (T, R, C, C): 2C + T + R = 22. Replace T + R by 14: 2C + 14 = 22, so C = 4.1 mark
- Row 3 (C, T, C, C): 3C + T = 18 ⇒ 12 + T = 18 ⇒ T = 6. Then R = 14 − 6 = 8.1 mark
- Row 2 (H, R, H, R): 2H + 2R = 30 ⇒ 2H + 16 = 30 ⇒ H = 7.½ mark
- Check the remaining totals: row 4: 3C + R = 12 + 8 = 20 ✓; column 2: 3R + T = 24 + 6 = 30 ✓; column 3: R + H + 2C = 8 + 7 + 8 = 23 ✓. Missing column 1: T + H + 2C = 6 + 7 + 8 = 21.1 mark
Check: Every given total is matched: 28, 30, 18, 20 (rows) and 30, 23, 22 (columns) ✓.
Answer to write in the exam
Row 1: 2T + 2R = 28 ⇒ T + R = 14
Column 4: 2C + T + R = 22 ⇒ 2C + 14 = 22 ⇒ C = 4
Row 3: 3C + T = 18 ⇒ T = 6; R = 14 − 6 = 8
Row 2: 2H + 2R = 30 ⇒ H = 7
Column 1: T + H + 2C = 6 + 7 + 8 = 21
∴ Circle 4, triangle 6, rectangle 8, hexagon 7; ? = 21
Common mistakes that cost marks
- Trying to find T and R separately from row 1 alone. One equation in two unknowns has many solutions; you need a second equation.
- Miscounting the shapes in a row, e.g. reading row 3 as 2C + T instead of 3C + T.
How this can come in the exam
In a puzzle, 2 stars + 1 moon = 13 and 1 star + 2 moons = 11. Find the value of a star and a moon, and of 3 stars + 3 moons.
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Add: 3(star + moon) = 24 ⇒ star + moon = 8, so 3 stars + 3 moons = 24 (1 mark). Subtract: star − moon = 2 (1 mark) ⇒ star = 5, moon = 3 (1 mark).Try one yourself
Square + square + triangle = 17 and square + triangle + triangle = 19. Find each.
Show answer
Add: 3(S + T) = 36 ⇒ S + T = 12; subtract: T − S = 2 ⇒ S = 5, T = 7.
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