How many squares does the diagonal pass through?
A hard-level "Count" problem.Counting crossings is the quick way. Just watch for the lattice points, where two lines are cleared in a single step — subtract those.
GivenA 6 by 4 grid of squares with one diagonal drawn across it.
Hints
- Shading and counting one by one is hard work. Instead ask when the square changes: the diagonal moves into a new square exactly when it crosses a grid line.
- Only the interior lines get crossed: 6 − 1 = 5 vertical and 4 − 1 = 3 horizontal, that is 8 lines.
- Start in the first square and gain one at every crossing: 1 + 8 = 9. Except that there is a catch.
- Look at the red dot: the diagonal passes exactly through a lattice point. It crosses a vertical and a horizontal line at the same moment, yet enters only one new square — so that is one crossing counted too many. Here it happens 1 time.
- So 9 − 1 = 8. Tidied up: width + height − their greatest common divisor. Shade them in and there are indeed 8.
Answer 8
More "Count" problems
- How many turns does the small circle make?
- How many cubes have 2 painted faces?
- Into how many pieces do 4 lines cut the paper?
- In how many ways can the three chosen rods form a triangle?
- How many triangles are there in the figure?
- How many squares are there altogether?