Problem 16 · 2018 Math Kangaroo
Medium
Spatial & Visual Reasoning
cube-viewscareful-counting
Tobias glues 10 cubes together to form the object shown. He paints all of it, even the bottom. How many of the cubes then have exactly 4 faces coloured?
Show answer
Answer: C — 8
Show hints
Hint 1 of 3
Every cube has 6 faces; the bottom is painted too, so a face stays unpainted only where it is glued to a neighbour cube.
Still stuck? Show hint 2 →
Hint 2 of 3
A cube shows exactly 4 painted faces when exactly 2 of its faces are glued to neighbours.
Still stuck? Show hint 3 →
Hint 3 of 3
The whole object is one long bent line of cubes, so most cubes touch a neighbour on each end.
Show solution
Approach: the cubes form one long bent chain; count how many touch a neighbour on exactly two sides
- All 6 faces of a cube get painted except the ones glued to a neighbour, so a cube shows 4 painted faces exactly when 2 faces are glued.
- The 10 cubes are joined into one long bending line (up the left arm, across the bottom, up the right side), so each cube in the middle of the line is glued to a neighbour on two sides, even the corner cubes where the line turns.
- Only the two cubes at the very ends of the line are glued on just one side, so they show 5 painted faces; the other 10 − 2 = 8 cubes each show exactly 4 painted faces, answer C.
Mark:
· log in to save