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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.

rectangular pizza with a circular hole

I couldn't resist sharing it with the MSE community. Enjoy!

psmears
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Robert Z
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    I'm reminded of the more general (but less constructive) [pancake theorem](https://en.wikipedia.org/wiki/Ham_sandwich_theorem#Two-dimensional_variant:_proof_using_a_rotating-knife). – Mark S. Sep 29 '17 at 11:48
  • This one's been around for a while. https://www.cartalk.com/content/cutting-holey-cake – Geoffrey Sep 29 '17 at 15:39
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    in germany this problem (or some variant of it) is quiet commonly asked in job interviews (for jobs with an 'analytic' background) – tired Sep 29 '17 at 17:14
  • Is the cut circle always entirely within the pizza? Or can the 'cut out' intersect the edge of the pizza, making the cutout less than a complete circle? – DJClayworth Sep 29 '17 at 17:29
  • @DJClayworth I think that the hole should be always entirely within the pizza, but this is not specified explicitly. – Robert Z Sep 29 '17 at 17:36
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    The nice answer is invalid if part of the circle is outside the pizza, so it must be that the circle is entirely inside the pizza! – alexis Sep 29 '17 at 19:11
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    Pizzas have some thickness, just cut it in half along the thin side. One person gets the top, the other gets the bottom. Completely fair /s – Justin Sep 30 '17 at 06:09
  • @Justin but the top part is heavier than the bottom part! – Zereges Sep 30 '17 at 10:38
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    The problem statement says that the chef "cuts out a circular piece". That seems to unambiguously imply that the cut cannot extend beyond the edge of the pizza, because then the cut would not produce a circular piece. – Tanner Swett Oct 13 '17 at 16:01
  • @alexis jor objection is wrong. if "part of the circle is outside the Pizza" then the Chief didn't cut out a circular Piece from the Pizza.but only a Segment of a circle. – miracle173 May 07 '18 at 11:04

4 Answers4

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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.

M. Winter
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    There's a nice physical approach to this too. A cut that splits the pizza evenly in two must pass through the pizza's center of mass. Assuming uniform mass distribution, symmetry tells us that the pizza's COM must be on a line that passes through the whole pizza's COM and the hole's center of "mass". – Dancrumb Sep 29 '17 at 13:54
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    @Dancrumb It is not the case that a cut which splits the pizza evenly in two must pass through the pizza's center of mass. The particular line given by this answer does, but others might not. – jwg Sep 29 '17 at 15:17
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    @jwg: Indeed most of the cuts of this holey pizza into two equal areas do not pass through the centre of mass. I suspect that except in special cases M. Winter's cut is the only one which does – Henry Sep 29 '17 at 15:24
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    A line that splits the pizza evenly in half must, by definition, have half of the pizza one one side and half on the other. This means half the mass is on one side and half is on the other. Thus, this is a line that you can balance the pizza on. Thus, the COM must be on that line. – Dancrumb Sep 29 '17 at 16:49
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    I would have counted this as two cuts, one on either side of the hole. @Dancrumb the center of mass of the un-holed pizza. –  Sep 29 '17 at 17:24
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    After reading around, I realize I am incorrect. I've failed to account for torque. A line through the COM may have a larger area on one side that is close to the line and a smaller area further away from the line that will balance, but not be of equal area. – Dancrumb Sep 29 '17 at 18:10
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    This answer is fantastically elegant. – Steven Stadnicki Sep 29 '17 at 18:26
  • Love this answer and wonder how it would modified for a pizza with two holes... – airstrike Sep 29 '17 at 21:43
  • @Dancrumb Having half of the mass on both sides of the line is not the same as the line passing through the COG. It's like the difference between median and arithmetic mean – Hagen von Eitzen Sep 30 '17 at 12:29
  • @AndreTerra: If the holes are exactly the same size, then we are done by the same argument. If not, ... good luck finding a quick solution before the pizza gets cold! – user21820 Oct 01 '17 at 17:03
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    @user21820 Most of the time the centers of the pizza and the two holes are not on the same line. How can you apply *the same argument*? – M. Winter Oct 01 '17 at 18:25
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    Why not mush the pizza into a ball, and then cut that in half? – will Oct 01 '17 at 22:34
  • @M.Winter: Two holes of the **same** size will **together** form a 180-degree rotationally symmetric shape about their **centre**. Your own post was all about using that fact. Didn't you see it? – user21820 Oct 02 '17 at 06:18
  • @user21820 Sorry I do not understand. Only because they are of the same size does not mean they are arranged symmetrically with respect to the pizzas center, right? – M. Winter Oct 02 '17 at 06:22
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    @M.Winter: Draw two circles of the same size on a piece of paper. Find their **joint centre**. Prove that the **joint shape** made up of the two circles is 180-degree rotationally symmetric about that joint centre. I'm sure you can get it if you think a bit more before replying. =) Also, once you get it, please delete your first reply to my because it is misleading to others but has upvotes. I will then delete my subsequent replies. Thanks. – user21820 Oct 02 '17 at 06:27
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    @user21820 Ah I see. Embarrassing :D Indeed using my very own argument. I thought with "my argument" you meant joining the centers of the circles and the rectangle. I am sorry about the upvotes to this other comment, but I think it should stay because it is the origin of this small discussion leading to a full explanation. – M. Winter Oct 02 '17 at 06:40
  • @M.Winter: Alright thanks for the response! =) – user21820 Oct 02 '17 at 06:41
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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
    |   \  \ /|
    +---/--/ \+
Zoltan K.
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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.

https://ggbm.at/P97VMYzX

Travis N. Miller
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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.

Coo
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