Problem 27 · 2021 Math Kangaroo
Stretch
Geometry & Measurement
Number Theory
spatial-reasoninggrid-counting
A 3×4×5 cuboid consists of 60 identical small cubes. A termite eats its way along the diagonal from P to Q. This diagonal does not intersect the edges of any small cube inside the cuboid. How many of the small cubes does it pass through on its journey?
Show answer
Answer: C — 10
Show hints
Hint 1 of 2
The diagonal of an a×b×c grid crosses a + b + c interior 'sheets', but crossings at shared edges count once.
Still stuck? Show hint 2 →
Hint 2 of 2
Use a + b + c minus the pairwise gcds plus the triple gcd.
Show solution
Approach: count unit cubes a space-diagonal passes through
- For a 3×4×5 box the count is a+b+c − gcd(a,b) − gcd(b,c) − gcd(a,c) + gcd(a,b,c).
- = 3+4+5 − 1 − 1 − 1 + 1 = 10.
- So the answer is C.
Mark:
· log in to save