# Cutting a cake into 8 pieces

Say, you are given a cake which you must share with 7 others. So, you must cut the cake into 8 pieces. But, you are only allowed to make 3 straight cuts. You cannot move pieces of the cake after the first cut.

Obviously, this doesn't work in the plane. So you need a 3D solution.

A 3D solution is simple. Cut the cake with 2 perpendicular cuts through the center, then make a horizontal cut at half the height of the cake. It is not fair regarding topping, but you have 8 pieces.

• Not fair regarding topping?! :O why bother Oct 12 '14 at 19:31
• Assume a perfectly spherical cake... Oct 16 '14 at 16:18
• I bet birthday cakes in the ISS have topings all around. Oct 16 '14 at 16:30
• @FlorianF Cakes in space come in bags and you eat them with a straw. Cutting the bag into 8 pieces is not advised. May 29 '15 at 20:18
• @Zibbobz A cylinder will do just fine. As will a cube. Mar 30 '18 at 18:13

Florian's answer mentions that it's 'obvious' that the cuts don't work in the plane, but I figure it's worth a short proof that you in fact can't cut a (convex!) 2d cake into eight pieces with three slices.

Firstly, since the cake is convex we may as well say that it's infinitely large and just look at cutting the plane into pieces; this won't affect the maximum number of pieces we can get. Now, since each slice is a line, any two of them intersect in at most one point. If all three of them intersect in the same point then they partition the plane into at most six regions. Likewise, if any two slices don't intersect, then they partition the plane into at most six regions. Otherwise, pick two of the slices: they divide the plane into four quadrants. Now, the third slice intersects the other two in one point each, and those two intersection points border a common quadrant. Whichever quadrant is opposite to that one can't be cut by the third slice, because either getting 'in' or 'out' of it would require another intersection with one of the first two slices definiing the quadrants; this means that the three slices partition the plane into at most 4 (the quadrants from the first two slices) + 3 (the new regions defined by the third slice) =7 pieces.

And why convex? Well, cutting this nonconvex 'cake' into eight pieces with just three slices is left as an exercise to the reader:

• who wants a 2D cake anyway. not at all filling. Dec 4 '15 at 4:13

As in the image; I will cut the cake horizontally first and two vertical cuts into it will be added.
So it can have 8 pieces in total finally without even moving the cake position.
I guess that's very simple and easy way. Isn't it? :D.

cut the cake in an even X making 4 slices. Then stack the slices on top of each other and cut them down the center making 8 equal pieces with equal icing

• "You cannot move pieces of the cake after the first cut. " Mar 17 '16 at 13:22

The following cake shapes can be cut into 8 pieces with 3 straight cuts: cone, cube, regular octahedron, torus, isosceles trapezoid, prolate ellipsoid, orthogonal parallelepiped, sphere

• I feel like this needs some sort of evidence to back up these claims, could you possibly provide some? Jun 3 '20 at 22:10
• CUBE: if you bisect the three sides, A, B, C, then you draw straight lines parallel to the faces of the cube, from the mid-points of the three sides, cutting the cube into eight equal parts. SPHERE: draw two diameters perpendicular to each other and then make a third cut through the equator, dividing the sphere into eight equal parts. Two examples should suffice. Jun 3 '20 at 23:24

It is possible to cut a (convex) cake into 8 identical pieces with 1 straight cut:

Cut 4 pieces at 45 degree angles - same as diving a wristwatch from 10:30-1:30 , 1:30-4:30, 4:30-7:30 and 7:30-10:30. Each piece is equal to 3 hours on a watch. Take the piece with 3 o'clock portion and align this with the 12 o'clock position. Similarly, align the 9 o'clock portion with 6 o'clock. Make sure all the center portions are aligned in one straight line which will pass through 3, 12, the cake center, 6 and 9. A single cut along this line will give 8 exact pieces equal to 1:30 hrs on a clock. This does not require stacking which can destroy the cake icing etc. Enjoy the perfect 8 equal pieces.

• Firstly, welcome to S.E. Puzzling. We hope you enjoy it here! Unfortunately, the OP specifiec that you cannot move pieces after you make the first cut, so this approach would not work. May 29 '15 at 20:22

Just for the fun, assume the cake is only 2 dimensional, so that the other answers don't work. A solution is still possible. Cut the cake one line through the center and now bend it to form a cone. Be careful not to move the pieces, as that is not allowed, merely bend them. Some parts will overlap especially on the bottom of the cone. Now cut the cone in a straight horizontal line about at half height. Unbend it and you have a circular cut but you only did a straight cut. Now just do a second cut through the center and you have 8 pieces and all with topping. Enjoy!

(the solution still works for a 3 dimensional cake. After all it works with paper and paper is 3 dimensional. Cake is a bit (a lot) thicker, so it might be a bit more difficult but it works in theory)

• Bending the cake would likely count as moving it (how else would you bend it other than moving parts of it) but interesting proposal Dec 14 '18 at 14:12
• Well strictly seen cutting is also bending until the material breaks. But the other two cuts don't count as moving, so bending is not moving. Dec 14 '18 at 14:30
• I think its one of those cases where a square is a rectangle but a rectangle isn't a square. Cutting is allowed and is technically up to a point bending, but bending on its own as a separate action from cutting would only be moving it and likely not allowed. Then again it is up to the creator but seeing as this question is over 4 years old now and already has an accepted answer its unlikely to get much of a response. Dec 14 '18 at 14:35

How about cutting the cake in an even X making 4 slices. then cutting a concentric circle making them 8 pieces.

• Welcome to puzzling.stackexchange! Thank you for contributing. It is customary to use ">!" to hide answers that contain spoilers. I have added these. Also, I believe the question specifies that the cuts are straight. Maybe you should re-think your answer. Keep puzzling! Mar 17 '16 at 13:28
• @HughMeyers This is now the only answer on this question to use spoilertags ... Jul 11 '17 at 16:01

It is simple.First of all find 360 degree divided by 8. So we get 45 degrees.Now, in our cake concept, cake is the circle with 360 degrees and as we cut into 8 equal parts, each cut part should have 45 degrees. Now how to do this is: Cut the cake directly into half.Take 45 degrees from an end of the cake to right side.Cut the cake right at this new position, in full to further half. Repeat this step once again. We get 8 equal parts of piece cakes.

• You're not allowed to move the pieces. Jul 11 '17 at 19:13