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Puzzle answers, explained

Five puzzles from our maths poster, solved step by step. Each answer uses a rule from the toolkit. Try each one first, then tap Show answer to watch it solved.

tried them yourself first? good!

01 · Hidden tile

EASY

The tiles go 144, 169, ?, 225. What number hides behind the ? tile?

144 169 ? 225
ANSWER & EXPLANATION
144 169 ? 196 225 12² 13² 14² 15²
  1. Check the squares table: 144 = 12 x 12 and 169 = 13 x 13.
  2. And the last tile: 225 = 15 x 15. So the tiles are squares of 12, 13, ?, 15.
  3. The missing one must be 14 x 14 = 196.
ANSWER 196

02 · Sweet share

EASY

Bags have 431, 515 and 828 sweets. Which bag can be shared equally among 9 friends?

431 515 828
ANSWER & EXPLANATION
431 515 828 4+3+1 = 8 ✗ 5+1+5 = 11 ✗ 8+2+8 = 18 ✓ 828 ÷ 9 = 92 each
  1. Toolkit rule: a number divides by 9 if its digit sum divides by 9.
  2. 431 → 8 and 515 → 11. Neither is a multiple of 9.
  3. 828 → 8 + 2 + 8 = 18, and 18 = 9 x 2. Each friend gets 92 sweets.
ANSWER 828

03 · The ladder

EASY

A ladder reaches 8 m up a wall, with its foot 6 m away. How long is the ladder?

8 m6 m?
ANSWER & EXPLANATION
8 m 6 m ? 10 m 36 6² 64 8² 36+64 = 100
  1. The wall, the ground and the ladder make a right-angled triangle. The ladder is the longest side, c.
  2. Pythagoras: a² + b² = c², so 6² + 8² = 36 + 64 = 100.
  3. c = √100 = 10 m. Shortcut: 6-8-10 is just the magic triple 3-4-5 doubled!
ANSWER 10 m

04 · Garden in 4 pieces

MEDIUM

A square garden of side 13 m is split into 4 plots (10 m + 3 m on each side). Find its area without doing 13 x 13.

10 m3 m 10 m3 m
ANSWER & EXPLANATION
10 m 3 m 10 m 3 m 100 30 30 9 (a + b)² = a² + 2ab + b² (10 + 3)² = 100 + 60 + 9 169 m²
  1. Big plot: 10 x 10 = 100 m².
  2. Two long strips: 10 x 3 = 30 m² each, so 60 m².
  3. Small corner: 3 x 3 = 9 m².
  4. Total = 100 + 60 + 9 = 169 m². That is exactly (a + b)² = a² + 2ab + b², drawn as a picture!
ANSWER 169 m²

05 · Melt the ball

HARD

A metal ball (radius 3 cm) is melted and poured into a can of the same radius. How tall is the metal in the can?

r = 3 cm r = 3 cmh = ?
ANSWER & EXPLANATION
3 cm ⁴⁄₃ x π x 3³ = 36π 4 cm r = 3 cm π x 3² x h = 9π x h h = 4 cm
  1. Melting does not change the amount of metal. So volume of ball = volume of metal in the can.
  2. Ball (sphere): ⁴⁄₃πr³ = ⁴⁄₃ x π x 27 = 36π cm³.
  3. Can (cylinder): πr²h = π x 9 x h = 9πh.
  4. 9πh = 36π, so h = 36 ÷ 9 = 4 cm. The π cancels out, no calculator needed!
ANSWER 4 cm
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