Into how many pieces do 4 lines cut the paper?
A hard-level "Count" problem.Do not try to count the regions. Add one line at a time and ask how many the count goes up by: it equals the number of pieces the new line is cut into.
GivenThere are 4 straight lines. Every two of them cross, and no three meet at one point.
Hints
- Start with no lines at all: the paper is still one piece. Now add lines one at a time and watch how many pieces are gained.
- One line splits the paper into two. That is one more piece.
- Line 2 meets each of the 1 existing lines exactly once (red dots). With 1 crossings, the new line is cut into 2 pieces — and it gains exactly that many regions. So +2, giving 4.
- Line 3 meets each of the 2 existing lines exactly once (red dots). With 2 crossings, the new line is cut into 3 pieces — and it gains exactly that many regions. So +3, giving 7.
- Line 4 meets each of the 3 existing lines exactly once (red dots). With 3 crossings, the new line is cut into 4 pieces — and it gains exactly that many regions. So +4, giving 11.
- Altogether 1 + 1 + 2 + … + 4 = 11. However many lines there are, the k-th line always adds k regions.
Answer 11
More "Count" problems
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- How many cubes have 2 painted faces?
- 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?
- How many squares does the diagonal pass through?