Noetic Problem of the Week

THE PROBLEM: 
What is the maximum number of pieces of pizza possible when it is cut with 5 straight line cuts (the lines are not necessary through the center of the pizza.)?

SOLUTION:  
The first cut
produce
s
1 extra piece: 1 + 1 = 2
.
The 2nd cut can at most go through 1 line and can
have at most
2 line
segments. So, the 2nd cut will at most produce 2 extra pieces
: 2 + 2 =
4.
The 3rd cut
can at most go through 2 lines
and can
have at most
3 line
segments. So, the 3rd cut can produce
at most
3 extra pieces
: 4 + 3 =
7
.
The 4th cut can at most go through 3 lines (the lines resulted from the
first 3 cuts) and can
have
at most 4 line segments. So, the 4th cut can
produces
at most
4 extra pieces: 7 + 4 = 11.
Based on the above reasoning, the 5th cut can at most go through 4 lines and can
have
at most 5 line segments. So, the 5th cut cat at most
produces 5 extra pieces:
11 + 5 = 16.
We can extend the above solution to any number of cuts. What is the maximum
n
umber of pieces of pizza possible when it is cut with 10 straight line cuts?
The first cut
produce
s
1 extra piece: 1 + 1 = 2
.
The 2nd cut can at most go through 1 line and can
have at most
2 line
segments. So, the 2nd cut will at most produce 2 extra pieces
: 2 + 2 =
4.
The 3rd cut
can at most go through 2 lines
and can
have at most
3 line
segments. So, the 3rd cut can produce
at most
3 extra pieces
: 4 + 3 =
7
.
The 4th cut can at most go through 3 lines (the lines resulted from the
first 3 cuts) and can
have
at most 4 line segments. So, the 4th cut can
produces
at most
4 extra pieces: 7 + 4 = 11.
Based on the above reasoning, the 5th cut can at most go through 4 lines and can
have
at most 5 line segments. So, the 5th cut cat at most
produces 5 extra pieces:
11 + 5 = 16.
We can extend the above solution to any number of cuts. What is the maximum
n
umber of pieces of pizza possible when it is cut with 10 straight line cuts?