Problem 25 · 2016 Math Kangaroo
Stretch
Spatial & Visual Reasoning
cube-viewsspatial-reasoning
A big cube is made of 64 small cubes. Exactly one of them is grey (see diagram). Two cubes are neighbours if they share a common face. On day one the grey cube colours all of its neighbours grey. On day two all grey cubes again colour all of their neighbours grey. How many of the 64 little cubes are grey at the end of the second day?
Show answer
Answer: E — 17
Show hints
Hint 1 of 3
After day one the grey cube and all cubes sharing a face with it are grey.
Still stuck? Show hint 2 →
Hint 2 of 3
After day two add every cube sharing a face with those, so grey reaches anything within 2 face-steps of the start.
Still stuck? Show hint 3 →
Hint 3 of 3
Count the cubes you can reach in at most 2 face-steps, remembering the block is only 4 by 4 by 4 so some directions run off the edge.
Show solution
Approach: count cubes reachable within two face-steps
- A cube ends up grey exactly when it can be reached from the start in at most two face-to-face steps (one step on day one, one on day two).
- The grey cube sits on the top face just in from the back, so day one greys its 5 face-neighbours, and day two greys the new cubes one more step out.
- Counting every cube within two face-steps (stopping at the outer faces of the 4 by 4 by 4 block) gives 17 grey cubes.
Mark:
· log in to save