In a recent issue of Crux, at the end of the editorial (which is public), it appears the following very nice problem by Peter Liljedahl.
I couldn't resist sharing it with the MSE community. Enjoy!
In a recent issue of Crux, at the end of the editorial (which is public), it appears the following very nice problem by Peter Liljedahl.
I couldn't resist sharing it with the MSE community. Enjoy!
Nice riddle! My solution would be to cut along a line through the center of the circle and the center of the rectangle.
Proof.
A cut through the center of a circle divides it into pices of equal size. The same holds for rectangles. Therefore everyone gets the same amount of pizza minus the same "amount of hole". $\square$
$\qquad$
It amazed me that this works for pizzas and holes of even stranger shapes as long as they are point-symmetric. In this way one can make the riddle even more interesting, e.g. an elliptic pizza with a hole in the shape of a 6-armed star.
What does it exactly mean "1 cut"? Does it mean a straight line, or that the knife is always held down, or it does not leave the premise of the pizza - is the hole within the premise of the pizza? etc...
Depending on the true meaning of "1 cut", other answers are possible, too, some of which can be used in a larger set of holes than the original question.
I lack the reps to add upload img, so here is an ascii art:
Hole on the right, zig-zag cutline in the middle, B has the hole, so a half/hole from A is cut away, and given to B:
+----|----+
| | |
| | _ |
A | / / \| B
| \ \_/|
+----|----+
Solution for an unorthodox hole, the hole (intersecting the perimeter or the pizza on the right, the zig-zag cut in the middle:
+----|----+
| | |
| | _ |
A | / / \| B
| \ \ /|
+---/--/ \+
I stumbled onto this problem and thought it would fit nicely as an activity in my classroom. I created a GeoGebra applet ofs this problem where students need to construct the midpoint and then measure the sides of their slices. When clicking the button it randomizes the pizza so students will be able to see if their method works for all of Hole in One Pizzas. I thought I would include the link here in case any other teachers came across this problem. It is a worksheet but you could just copy the applet.
Lay both pizza's one on top of the other and cut through the whole so that it sliced in half. Each person get a slice from the pizza with the whole and the remainder of the other pizza. So the exterior area of the pizzas are equal and the share of the whole is equal.