Plane Cutting
Using 20 straight cuts, what is the maximum number of pieces that a paper can be divided into?
Note: You are not allowed to fold, tear or crease the paper.
The answer is 211.
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2 solutions
I did the same math, different logic.
Number of pieces = number of lines + number of intersections + 1 To maximize the number of intersections, each new line crosses each existing line once. The 1st line crosses 0, the 2nd crosses 1, the 3rd crosses 2, etc. N = 2 0 + ( 0 + 1 + 2 + 3 + 4 + ⋯ + 1 8 + 1 9 ) + 1
N = 2 0 + ( 2 2 0 ( 1 9 + 0 ) ) + 1
N = 2 0 + 1 9 0 + 1 = 2 1 1
Excellent problem, but it takes a mathematician to know that after a cut the pieces stay in place no matter how many there are. More common folk would take the two pieces after the first cut, put them on top of each other (not prohibited by the wording) and cut through both of them. If one were to continue in that vein, the result would be 2^20.
The general formula is 2 n 2 + n + 2 Put n = 2 0 to get the answer as 2 1 1 .
Each time you make a slice, you can slice through each of the cuts you have already made.
So you start with one piece. The first slice adds one piece. The second slice adds 2 pieces. And so on.
So, the total number of pieces after 2 0 cuts will be, N = 1 + 1 + 2 + 3 + 4 + 5 + . . . . + 2 0 .
N = 1 + i = 1 ∑ 2 0 i = 1 + 2 2 0 ( 2 0 + 1 ) = 2 1 1