+4622 With two straight cuts, we can divide a flat pancake into four pieces:
https://latex.artofproblemsolving.com/9/0/9/909e28e4df779da2cd663ff4a1bddfae59ee7732.png
What is the fewest number of straight cuts that will divide a flat pancake into 16 pieces? (The pieces do not need to be the same shape or size.)
+9481 Well, this probably isn't the professional way to do it...but I think the answer is 5 .
+2511 Actually, that is the professional way of doing it, when you’re teaching concepts.
Here’s a LaTex formatted, reproduction from
http://mathworld.wolfram.com/CircleDivisionbyLines.html
deriving the general solution formula for circle cutting (pancake cutting) problems in a plane.
\(\begin{array}{lrl} F(1) &=& 2 \\ F(2) &=& 2+f(1) \\ F(n) &=&n+f(n-1) \\ \small \text{Therefore, }\\ &=&n+(n-1)+f(n-2)\\ &=&f(1)+\sum \limits_{k=2}^{n}k \\ &=&2+\dfrac{1}{2}(n+2)(n-1) \\ &=&\dfrac{1}{2}(n^2+n+2) \\ \end {array} \)
Here’s the specific solution for this problem using the quadratic formula. As expected, this matches Hectictar’s solution.
\(\dfrac{(n^2+n+2)}{2}=16\\ n^2+n+2=32\\ n^2+n-30=0\\ \dfrac{-1+\sqrt{1^2-4\cdot \:1(-30)}}{2 \cdot 1} = 5\\ \dfrac{-1-\sqrt{1^2-4\cdot \:1(-30)}}{2\cdot \:1} = -6 \small \text{ (Not used as a solution for this problem in this universe.* }\\ \)
---------------
*The negative six (-6) is a solution in an alternate universe.
I’ve visited that universe. I noticed on the forum there, that Sisyphus is the prolific solution master for mathematics, and CPhill is the well-known rock-roller.
GA
