Problem 18 · 2013 Math Kangaroo
Medium
Geometry & Measurement
Number Theory
grid-countingspatial-reasoning
In the 8×6 grid pictured, there are 24 squares that are not cut by either of the two diagonals. Now we draw the two diagonals on a 10×6 grid. How many squares of this grid will not be cut by either diagonal?
Show answer
Answer: E — 32
Show hints
Hint 1 of 2
A diagonal of an m×n grid passes through m + n − gcd(m,n) unit squares.
Still stuck? Show hint 2 →
Hint 2 of 2
Subtract the squares both diagonals touch from the total to get the uncut ones.
Show solution
Approach: count squares a diagonal crosses
- One diagonal of a 10×6 grid crosses 10 + 6 − gcd(10,6) = 14 squares.
- Both diagonals meet at the centre lattice point and share no cut square, so together they cut 14 + 14 = 28.
- The grid has 60 squares, so 60 − 28 = 32 are uncut.
- So 32 squares are not cut.
Mark:
· log in to save