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Can you cut a pizza (circle) into 12 congruent pieces, such that half of them have crust (circle boundary), while the other half do not? The pieces must have the same shape and area, but can be mirrors of each other.

Bonus: Can you do it with identical pieces, where pieces are not mirrors of each other?

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    $\begingroup$ For the record I like my pizza with crust :) $\endgroup$
    – Dmitry Kamenetsky
    Commented Oct 8, 2021 at 13:21
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    $\begingroup$ I've always loved this question. There are in fact many solutions, but I'll let others have a go at it before I post an answer. Do you allow answers where pieces have one point on the boundary? $\endgroup$
    – Jaap Scherphuis
    Commented Oct 8, 2021 at 13:57
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    $\begingroup$ I dare you to put this into your "additional instructions" when ordering a pizza. $\endgroup$
    – Ian MacDonald
    Commented Oct 8, 2021 at 18:08
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    $\begingroup$ By any chance, did you read "Things to make and do in the 4th dimension" by Matt Parker? ;) $\endgroup$
    – QBrute
    Commented Oct 9, 2021 at 12:14
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    $\begingroup$ Here is the same question from math stack exchange that may be of interest (although without the specification of 12 pieces): math.stackexchange.com/questions/481527/… $\endgroup$
    – Carmeister
    Commented Oct 10, 2021 at 2:40

3 Answers 3

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This paper by Joel Haddley and Stephen Worsley answers a slightly different question - finding monohedral disc dissections where not all pieces touch the centre - but the results generally apply to this problem too.

My favourite answer is this one:

enter image description here

Note that this one has interior pieces that don't even have a single point on the boundary.

There is an infinite family of solutions, that interpolates between sybog64's solution and the one above:

enter image description here

The paper also shows some neat variations of loopy walt's solution:

enter image description here

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  • $\begingroup$ Neat, indeed... $\endgroup$
    – loopy walt
    Commented Oct 9, 2021 at 9:18
  • $\begingroup$ Love these constructions. It's great to see that interior pieces can have no points on the boundary. $\endgroup$
    – Dmitry Kamenetsky
    Commented Oct 9, 2021 at 10:09
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    $\begingroup$ Some of those are going to be pretty tricky to slice and serve! $\endgroup$
    – Toby Speight
    Commented Oct 9, 2021 at 10:25
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    $\begingroup$ This might be very useful for gerrymandering… $\endgroup$
    – Aganju
    Commented Oct 9, 2021 at 14:27
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This is a minor upgrade on @sybog64's answer:

enter image description here

One way of thinking about it is to start with this

enter image description here

configuration and then taking groups of 2 slices and rotating each group by 120°.

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    $\begingroup$ @JoelRondeau yes, exactly. $\endgroup$
    – loopy walt
    Commented Oct 8, 2021 at 19:50
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    $\begingroup$ So assuming I have simple kitchen things, how do I slice my pizza thus? The radial cuts through the center are easy enough (butcher's twine to find the center of the pizza and get the radius. Pick a point on the edge to be a pivot for the folded string, and mark an arc with a pizza cutter from the crust to the center of the pizza. Move the string's pivot to where the crust edge was cut, and repeat until 6 arcs are cut). I don't know how to easily find the other pivot points, though. $\endgroup$
    – Dewi Morgan
    Commented Oct 8, 2021 at 23:36
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    $\begingroup$ @DewiMorgan I'm not fluent with advanced kitchen implements but geometrically the circle centres for the remaing 6 arcs can be obtained by rotating the centre of the pizza 30° around each of the corners of the large hexagon. $\endgroup$
    – loopy walt
    Commented Oct 9, 2021 at 0:06
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    $\begingroup$ @loopywalt "then taking groups of 2 slices and rotating each group by 120°" I prefer to think of it as mirroring them, as two pieces form a symmetrical shield shape. With this point of view you don't have to do all 6 pairs at the same time. $\endgroup$
    – Jaap Scherphuis
    Commented Oct 9, 2021 at 8:13
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    $\begingroup$ @JikkuJose shapely.readthedocs.io/en/stable/index.html $\endgroup$
    – loopy walt
    Commented Oct 9, 2021 at 17:00
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I haven't found a perfect solution, my pieces are symmetrical but not identical

layout of the cut pizza

the pizza is first cut in 6 pieces following an arcs centered on the corner of the last cut piece with radius same as the pizza. the 6 pieces are then cut along their axis of symmetry to get one crusty and one crustless piece

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  • $\begingroup$ Oh, that's a good point. I hadn't realised that identical probably means that mirror images are not allowed. I do know of just one solution without mirror images though. $\endgroup$
    – Jaap Scherphuis
    Commented Oct 8, 2021 at 14:48
  • $\begingroup$ Somehow I like this answer more than the other (which is more "correct") ... $\endgroup$
    – WhatsUp
    Commented Oct 8, 2021 at 22:41
  • $\begingroup$ This is correct and well done! $\endgroup$
    – Dmitry Kamenetsky
    Commented Oct 8, 2021 at 22:54
  • $\begingroup$ I got the puzzle from here, which shows an actual pizza sliced in this way: news.liverpool.ac.uk/2016/01/12/… $\endgroup$
    – Dmitry Kamenetsky
    Commented Oct 8, 2021 at 22:56

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