Problem 28 · 2020 Math Kangaroo
Stretch
Spatial & Visual Reasoning
Counting & Probability
cube-viewsspatial-reasoningcareful-counting
Cleuza assembled the 2×2×2 block of equal balls shown beside, using one drop of glue at each contact point between two balls, for a total of 12 drops. She then glued on more balls until she completed a 4×3×2 block. How many extra drops of glue did she use?
Show answer
Answer: C — 34
Show hints
Hint 1 of 2
A glue drop sits at every place two balls touch face-to-face; count contacts in the finished 4×3×2 block.
Still stuck? Show hint 2 →
Hint 2 of 2
Subtract the 12 drops already used on the 2×2×2 block to get the extra drops.
Show solution
Approach: count touching pairs in the block
- Touching pairs in an a×b×c stack number (a−1)bc + a(b−1)c + ab(c−1).
- For 4×3×2 this is 3·6 + 4·2·2 + 4·3·1 = 18 + 16 + 12 = 46 drops.
- She already used 12 on the 2×2×2 block, so the extra is 46 − 12 = 34.
- The answer is 34, choice C.
Mark:
· log in to save