On Friday I set this puzzle….

You face a room of very hungry people.  A large pizza arrives and you are handed a knife and asked to cut the pizza into as many pieces as possible using just three straight cuts.  Because the pizza is very hot you cannot move any of the pieces (e.g. stacking them on top of one another and then cutting).  What is the maximum number of pieces that you can create using just three straight cuts?

If you have not tried to solve it, have a go now.  For everyone else, the answer is after the break…

The maximum is 7….

Unless you have come up with a more sneaky solution!  Did you?

I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for the Kindle (UK here and USA here) and on the iBookstore (UK here in the USA here). You can try 101 of the puzzles for free here.

1. Clair says:

Well, that’s perked my Monday morning right up. Quite excited that I got it right!

2. Neal Robins says:

If you cut twice through the middle and then effective chop through each slice (leaving half the base without topping) you can get eight pieces… although I don’t think anyone would want that solution!!

1. M says:

I thought of this too. But then I want a top piece.

2. AMWhy says:

Ditto. 8 slices.

3. Berhard says:

ditto… change the directions of the cuts, and 7 of 8 slices have at least a topping (but are very different in size… )

4. Ditto – precision cutting required!

5. Henry says:

If the pizza is too hot to touch, I’m not sure how you are going to pull off this particular trick.

3. Holden Caulfield says:

Either that was the easiest one ever or I just became super intelligent! I spent more time rethinking my solution for this one than I spend working out the other answers, convinced I must have it wrong as it was too obvious.

1. Carl says:

Yup. Exactly. Me too.

2. Waldo says:

Me three.

3. Joe says:

Me four, and I got the 3d solution too. (the 8 pieces one)

4. Michael Sternberg says:

Yup, that’s about it: for straight cuts max. 7 pieces in 2d (each line can intersect each other only once), and 8 in 3d only (each cut doubling the number of pieces).

I’m afraid there’s no way to make 7 equal-area pieces in 2d, but 6 is possible, with an H-shaped cutting pattern and a judicious choice for the “vertical” strokes.

1. Michael Sternberg says:

… and of course a classic evenly spaced triple cut through the center makes 6 fine congruent sectors just as well–no need to over-think it.

5. Ian Reeve says:

Saw a similar puzzle once featuring a ring doughnut, but can’t remember what the answer was.

1. Michael Sternberg says:

10, it would seem.

6. Morgan says:

You can get 8 by cutting into 4, then a horizontal cut through the pizza, side to side, so cutting each piece in half. It’s not orthodox, but then this is a maths question, not a cooking blog.

7. spiderabc1 says:

I did the same as number 2 here. Never thought about missing the centre point. Sure it’s different to the cake. (Not sure how healthy my Monday breakfast will be now.)

8. Bearing in mind the pizza is too hot to handle, how are all these people suggesting a horizontal cut through the pizza actually going to cut it? Some kind of samurai sword, maybe?

9. richard says:

how can you prove that you can’t cut the pizza so all 7 pieces are the same size

1. pac0san says:

It’s a classic problem. The solution is equivalent to the quadrature of the circle.

You can obtain 7 slices quite similar in area, but never equals.

2. Savant says:

An assertion is NOT a proof. Let’s see the proof.

On the face of it, I cannot see why it’s impossible to get the 7 pieces to have identical areas. I can see that it COULD be impossible but that’s not enough.

3. Michael Sternberg says:

The first cut must divide a circle of unit area into two circular segments with area A = 3/7 and 4/7, respectively (lemon an white in the referenced sketch). This determines the orthogonal distance r from the center of the circle to the secant.

The second cut is also pinned to the same distance r from the center. The only parameter is the angle α of intersection with cut 1. This angle is uniquely determined by the fact that the smallest wedge-shaped segment must have area 1/7.

Finally, the third cut, again at distance r from the center will complete the picture, producing 2 additional wedges at area 1/7.

The central triangle must have, by construction and mutual dependency, three equal angles α and therefore is equilateral with an incircle of radius r. Hence, it’s area are function of r is known, yet also must be 1/7.

Given A = 3/7, the value r as function of the circle’s radius R or area (1) is the solution of a transcendental equation, eq. (17) at Mathworld. After plugging in all the numbers, my hunch is that the resulting area of the inner triangle is not 1/7. Actual numerics or clever inequalities are left as an exercise to the reader 🙂

10. jamie says:

Yes, I got it too by drawing a circle and playing around with drawing chords through it. The best part of the answer is now I also have a solution for my kid who doesn’t like the crust edge of the pizza 🙂

11. mittfh says:

Yup, same solution I got. Three big slices, three medium slices, and a small slice. Although as there’d be no ‘edge’ to the middle slice, six would be more practical.

It would be interesting calculating the maximum number of slices with different numbers of cuts. Obviously one cut can only result in two slices, and two cuts four, so what about four / five cuts?

I’ve just cut 7 equal size pieces. Nobody complained.

13. Lazy T says:

I’m a bit disappointed there is no twist to the puzzle, but I enjoyed the food aspect. Do you know any puzzles involving a plateful of spaghetti?

1. Michael Sternberg says:

2. From the world of roadie-ing for a band: if a cable has one end, then it has another.

3. Lazy T says:

Dear Intertubes,
please send me a spaghetti pizza, please don’t cut it I’ll do that myself
ta
t

14. Peter Huesken says:

Imho. opinion 12 pieces is possible…

First we fold the whole pizza in half
(the restriction mentioned that you cannot move any of the pieces, but we ´re not suddenly calling the whole pizza a piece, right?)

Then we cut horizontally through the pizza (and do this very carefully around the bend), leaving half the base without topping.

Next we chop through the middle of the folded pizza

Finally we chop this result of the abovementioned steps in two.

15. otmshank says:

10 pieces without folding the pizza:

And then “chop through each slice (leaving half the base without topping) ” as explained above.

1. Michael Sternberg says:

If you’re leaving the domain of straight lines, why turn only once?

You could zig-zag across any diameter as often as you like, then repeat the cut pattern mirrored about that diameter. Bingo – as many pieces as you wish – to infinity and beyond, not even needing a third cut.

16. Camel says:

17. wagdog says:

A possible twist in the puzzle is that nowhere is it stated that the”large pizza” is actually round, or even convex.

http://www.favoritebrandrecipes.com/Recipes/065/4835001065.htm

Arrange to have a non-convex pizza be cooked and delivered, and you can end up with as many pieces as you want after three straight cuts.

18. Jerry says:

Yep. Got it.

19. Ok, you can’t move the slices. But if you wrap the entire pizza around a stick several times before slicing, you can get as many pieces as you want.

1. Lazy T says:

Champion !

20. Anonymous says:

Cut a circle, say, half way between the edge and the centre – then cut along the diameter twice – eight pieces…

21. Anonymous says:

get a cut wheel and have some

22. 8 if you’re allowed a horizontal cut (even if the pizza is with a bit of effort you won’t have to touch it)

23. Mark says:

I got an infinite number of pieces. No one said I had to lift the knife up so I just kept it down and started walking. Once I’d walked around the earth once I was able to cut the pizza again without lifting the knife. Perhaps my definition of a straight line on the surface of an oblate spheroid needs work… 😦

24. Sarah E. says:

What is it if you are only allowed to cut 6 straight cuts???