[puzzler] cutting french toast

[FlameBlade]

One claimed that he was able to form 12 rectangles just with 3 cuts of a french toast.

He's wrong. Why?

[Kyle]

Because rectangles are flat 2-dimensional objects and french toast exists in 3 dimensions?

Just off the top of my head, I can only think of a way to create 8.

[Mike]

My first answer is: "Because of the rounded corners of the bread."

But then taking it seriously... I'm making the assumption that our original piece of French toast is a rectangle. I'm also making the assumption that these are straight cuts. In that case, I don't see how three straight cuts can divide a convex shape into more than seven pieces total, much less rectangular pieces.

I can do it if the cuts don't have to be straight. Also, I can do it if I fold the toast before cutting so that a "single" cut actually slices the bread in two (or more) places.

I may be missing some basic assumptions, though.

[Kyle]

I stood the toast upright and sliced it again to make two slices that are half as thin. Then I lay them back down and did a vertical and a horizontal cut. Thus, eight pieces- four stacked on four.

[Eliahad]

If I get one curved cut, I can get 10.

[Zen]

If the toast is sufficiently thick and the knife sufficiently sharp and the bread actually square to begin with, I can get twelve rectangles with 1 cut.

[FlameBlade]

[Eliahad]

*whistles*

Just realized we're counting pieces, and Flame is counting rectangles. So um, we're in trouble.

[Stan]

If you bend or roll up up the french toast and then make a slice, does that count as one cut? If so you get 2-4 times more cuts per slice.

[poorpete]

My thought:
, but I assume I'm overlooking or misreading something.

[Stan]

If the goal is to get exactly 12, one solution is to wind up with 4 triangular pieces, each with 3 rectangles along their sides. It's really hard to get only 12 - if you cut into rectangular pieces, you get 6 rectangles per piece. A cut that is not perpendicular to the top would also help reduce the number rectangles on the sides.

[FlameBlade]

[poorpete]

Thanks. I guess you aren't counting this

[Stan]

If you're counting pieces, and if folding is not allowed, I think each cut can, at most, double the number of pieces you have. With 3 cuts, all perpendicular to each other, the most you can get is 2 to the 3rd or 8 pieces (or 48 rectangles).

[Phoebe]

If you can lift it up and fold it and stuff, why can't you rearrange pieces before cutting?

[Mike]

No one said we CAN fold and such... those have all been hypothetical and assumed to be against the rules. Rearranging pieces between cuts still ends up with eight pieces using straight cuts.

[Kyle]

Um... can I just say that it's really cool that we have our own mathematician on the site who comes up with puzzlers?

You're the best, Flame.

[Phoebe]

[Phoebe]

Same principle can be achieved if you just fold it, assuming you're allowed to fold parts between your cuts as well.

I'm assuming this is obvious but in case not let me explain. Just fold the toast over the first time, cut the third near the fold off so that you have three pieces. Fold the other direction, do the same thing again so that you have 9 pieces, and then just cut across the unfolded toast in one place to turn the nine into 12. No one said we could fold but no one said we couldn't fold so I'm just experimenting with my toast.

If it's illegitimate to fold that many pieces over the second time, but we can make an initial fold, you can just fold the whole thing up by thirds into a 3-layer roll with one edge folded under only part way. Cut off the long edge pieces so you separate a single piece on one edge, three in the middle, and two on the other (one fold-piece and one straight end) and then cut across those crossways.

[FlameBlade]

Big spoiler hint coming.