# Piece of cake

I asserted that the cuts can end in EACH OTHER, which is what you're proposing. But no ONE cut can both start AND end at an edge.
Sure they can. It's just that end up with a stupid solution. What you MEANT is probably one of:

- you forgot to stipulate that both cuts must be a necessary part of the solution.

OR

- *both* cuts cannot start AND end on edges.

Otherwise, imagine the following:
Code:
Cake with piece cut out:
+---------------------------------+
|                                 |
|             +-----+             |
|             |     |             |
|             +-----+             |
|                                 |
|                                 |
|                                 |
|                                 |
|                                 |
|                                 |
+---------------------------------+

Cut number 1:
+---------------------------------+
|                                 |
|             +-----+             |
|             |     |             |
|             +-----+             |
|                                 |
|                             ,---+
|                           ,'    |
|                      --,  |     |
|                        --'     |
|                                 |
+---------------------------------+

Cut number 2:
+----------------+----------------+
|                |                |
|             +--+--+             |
|             |     |             |
|             +--+--+             |
|                |                |
|                |            ,---+
|                |          ,'    |
|                |     --,  |     |
|                |       --'     |
|                |                |
+----------------+----------------+
Personally, I find this problem to be so ridiculously ambiguous that it's not worth bothering without further clarification. The problem that I want a solution for, which nobody can seem to find is:

---------------

A rectangle R exists. A smaller rectangle, P is completely contained within R. All of the edges of P are parallel to the edges of R. None of the edges of P overlap the edges of R. Shape K is defined as the subtraction of P from R.

Two straight line segments, C1 and C2 shall be made. These line segments may result in dividing K into multiple subsequent polygons. Devise a method for placing C1 and C2 such that the sum of the areas of some of the resulting polygons is exactly half of the original area of K.

Restrictions: At least one of C1's endpoints must be an already existing endpoint of one of the edges of K. C1 may be drawn parallel to an existing edge. If C1 is not drawn parallel to an existing edge, then both of its endpoints must be already existing endpoints on the edges of K.

Similarly, at least one of C2's endpoints must be an already existing endpoint of one of the edges of K, or the intersection of C1 and an existing edge. C2 may be drawn parallel to an existing edge of K, or parallel or perpendicular to C1. If C2 is not drawn parallel to an existing edge, or parallel or perpendicular to C1, then both of its endpoints must be already existing endpoints on the edges of K, or the intersections of C1 and an existing edge.

--------------------

That's what I'm assuming the question is asking. Otherwise, I can probably invent a slew of practical answers that are effectively cheats.

DaveE

DaveC426913
Gold Member
Sure they can. It's just that end up with a stupid solution.

...

What you MEANT is probably one of:

- you forgot to stipulate that both cuts must be a necessary part of the solution.

OR

- *both* cuts cannot start AND end on edges.

...

I can probably invent a slew of practical answers that are effectively cheats.

DaveE
Yes. So, since YOU are - like the rest of us - assuming that the spirit of the question presumes (without having to explictly state) that both cuts must be part of the solution...

AND that to do otherwise constitutes a cheat...

(Though it occurs to me (from your example) that nowhere does it stipulate that the cuts must be straight)

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You can do it in one cut, but in two cuts here is how it goes.

1 cut, cut it in half horizontally.
2 cut, cut it in half vertically.

Just switch the top pieces and you have 2 pieces with equal volumes.

>_M, I should be studying for the science & engineering physics exam on Wednesday haha.

that or slice through only half of the cake on the vertical or horizontal side with each cut.

doesn't end on an edge ;)

Yes. So, since YOU are - like the rest of us - assuming that the spirit of the question presumes (without having to explictly state) that both cuts must be part of the solution...

AND that to do otherwise constitutes a cheat...

Actually, I think it's grossly warranted, but I'm anal about these things :)

Mostly because people frequently make these sorts of assumptions without explicitly stating them-- and then people post stupid or repeated answers because they *don't* make those same assumptions, and they assume that they're correct. Which they *are*, because someone was lazy in not fully posting their assumptions along with the question.

(Though it occurs to me (from your example) that nowhere does it stipulate that the cuts must be straight)
If the lines are curved, I'd say that involves "measuring" and/or "calculating", which were expressly forbidden. I'd say that because I assume that humans are reasonably capable of making straight cuts pretty accurately, but NOT capable of making accurate curved cuts.

Admittedly, I'm also making an assumption about the knife used to cut. If you had a knife shaped like a curly-que, you'd have a much easier time cutting the cake in a circular pattern with no measuring at all. For that matter, maybe you're not using a knife to cut the cake, but a piece of fishing line or a laser.

But these assumptions seem outlandish to me, so in my stricter restatement of the problem, I explicitly specified that the line segments had to be straight.

DaveE

DaveC426913
Gold Member
Don't misunderstand me, I agree. Most problems of this nature depend on our assumptions - the classic http://www.dcu.ie/ctyi/puzzles/general/9dotpuz.htm" [Broken] being a textbook example.

But in this case, the problem states explicitly "Using two separate cuts, divide the cake into two equal (volume) pieces."

That the cuts must be straight is a big assumption, and fair game.

But that the cuts must be used in the solution I think is really skirting the edge of lawyering.

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DaveC426913
Gold Member
that or slice through only half of the cake on the vertical or horizontal side with each cut.

doesn't end on an edge ;)
No, but each cut ends at the other cut. Which was one of the possibilities I allowed.

But in this case, the problem states explicitly "Using two separate cuts, divide the cake into two equal (volume) pieces."
Hmm... I guess that's true-- I hadn't really considered that the language of "using" implies that the cuts must be necessary edges of the solution, but now that I think about it, I think you're right that that's implied.

DaveE

Your Thinking in the Wrong Plane

Imagine the width of the cake is X, the height is Y and the depth is Z. The rectangular slice is created by making two cuts, one across the X axis and one across the Y axis, cutting all the way through the cake in the Z direction. Now imagine cutting through the entire cake not along the XY Plane but rather halfway down the side of the cake in the Z plane, bisecting the cake so that half of the cake is still resting on the plate and the other half is now sitting on top of it. It won't matter what cuts are made in the original cake because all cuts go all the way through to the bottom of the cake, so cutting it in the Z plane will always work.

DaveC426913
Gold Member
Imagine the width of the cake is X, the height is Y and the depth is Z. The rectangular slice is created by making two cuts, one across the X axis and one across the Y axis, cutting all the way through the cake in the Z direction. Now imagine cutting through the entire cake not along the XY Plane but rather halfway down the side of the cake in the Z plane, bisecting the cake so that half of the cake is still resting on the plate and the other half is now sitting on top of it. It won't matter what cuts are made in the original cake because all cuts go all the way through to the bottom of the cake, so cutting it in the Z plane will always work.
Yes, that has been explored (you skipped to the end, didn't you?)

However, that's only one cut.

Cant you have one cut from an edge to a hole, and the other cut from the hole to another edge such that the volumes are equal? The assumption for this to work being that the hole is symmetrical so the volumes divided are also equal.

Cant you have one cut from an edge to a hole, and the other cut from the hole to another edge such that the volumes are equal? The assumption for this to work being that the hole is symmetrical so the volumes divided are also equal.
Nope. That involves measuring and/or the assumption that the hole is correctly positioned symetrically in the cake. The cake might look like this, for instance:

Code:
+----------------------+
|                      |
| +--+                 |
| |  |                 |
| |  |                 |
| |  |                 |
| +--+                 |
|                      |
|                      |
|                      |
|                      |
|                      |
|                      |
+----------------------+
Of course, there exist an infinite set of two cuts that can each be made from the edge to the hole such that you create pieces of cake that have equal volume. But in order to find them, you have to measure, which has been expressly forbidden. So any cuts you make have to have guidelines like using existing vertices. Personally, I'd say you could also make cuts that were parallel or perpendicular to existing edges (or cuts), and that that wouldn't constitute measuring, but for all we know, that's out too.

DaveE

cut the cake in half then cut the half without the hole in it in half. You end with two pieces of equal volume. You can eat the remainder.

Just to clarify:
1. We MUST make 2 cut?
2. We MUST end with only two, equal "volume" pieces?
3. We CANNOT measure anything but the final volumes?

And my question:

Do we know WHERE the missing cake segment IS? Are we given a blueprint of the cake and then have to figure it out, or is all we know that it has it?

My solution : Smash the cake flat, cut down the middle(2 cuts, from center to edge, then center to other edge). Who cares about density, the question is about volume. Smashing it takes away the missing part.

Also, how are you defining volume? The amount the cake would displace water if submerged? Or does a cake with a big missing part have the same volume as a cake with no missing part. Is the missing inside-part considered negative to the total volume?

Code:
+-----------+----------+
|           |          |
| +--+      |          |
| |  |      |          |
| |  |      |          |
| |  |      |          |
| +--+      |          |
|           |          |
|           |          |
|            \         |
|             \        |
|              \       |
|               \      |
+----------------------+

Ah, how about answering "It doesn't matter how you cut it, the Volume of a rectangle is always 0."

q3snt
I think you guys are getting way off track. I'm pretty sure out of whack had the right solution from the start. Not only did it meet all the requirements, but it was elegant just like riddle solutions are supposed to be.

I'm pretty sure out of whack had the right solution from the start. Not only did it meet all the requirements, but it was elegant just like riddle solutions are supposed to be.
I guess I just see that as violating the "No measuring" rule. It is a possible interpretation given that all you'd need is a long enough straight edge (no demarcations necessary) and the ability to make marks on the cake without technically making a cut. But I guess I just see that as "measuring".

But the other oddity I see with that solution is why bother stating that the rectangular slice was taken out with sides parallel to the edges of the cake? If the solution is Out Of Whack's, then it wouldn't matter what the orientation of the slice was, so long as it was bounded by the edges of the cake.

I guess my honest thought is that OOW's solution probably WAS the one the interviewers were looking for, and that the restatement of the problem here is incorrect, considering that jim.nastics was asked the question a long time ago, and didn't have the correct answer. Since he wasn't able to get a solution, it's entirely possible (and even probable) that he just didn't understand the wording of the question, and assumed "no measuring", or assumed parallel sides, or assumed that a line through both rectangle centers represented only a single cut.

But regardless-- I'd still be interested in knowing if there's an answer given the constraints as I posted earlier, even if it's technically a different problem all together.

DaveE

Throw the cake up high without it spinning and have it fall on the knife. The two parts that are cut will be equal right (like the cut will be through the COM)?