Solving the Two-Cut Rectangular Cake Puzzle

[jim.nastics]

Warning: I do not know the answer to this one. If this will keep you up at night, do not read on. I was asked this in a job interview a long time ago. They did like my answers, even though they were "wrong".

Still here? OK.

You are presented with a rectangular cake.

Somewhere at an arbitrary location in this rectangular cake, an arbitrarily sized rectangle of cake is missing.

The missing rectangle does not touch the edge, though it may be very close to it. In other words, the cake is not missing a corner.

Using two separate cuts, divide the cake into two equal (volume) pieces.

Wrong solution #1 (hidden text):
Using one cut, slice the cake horizontally, the top half is the same volume as the bottom half.

Wrong solution #2 (hidden text):
Use two knives. Balance the cake on one knife, then cut once along the line of balance using the second knife. Each half has the same volume. Note that there are an infinite number of solutions with this method


What are your thoughts? Every two-cut method I come up with has a case that does not work.

 

[DyslexicHobo]

The first solution that came to mind was your wrong solution #1. It's kind of like the trick where you go up to a jock with a telephone book and say "I bet you 20$ that i can rip this telephone book in half faster than you can".

 

[michealsmith]

yeah i have heard something like this before ...its a math trick i think something to do with when u bisect a rectangle you always end up halfing it ... or something like that ,,,well something that makes sense ...look into mathematics of rectangles you would find something there . ...I THINK

 

[chaoseverlasting]

Are there any constraints on the rectangular slice cut out? Is it in the same plane (are the planes parallel to the planes of the cake)? Can we use symmetry (is the damn slice symmetrical)?

 

[chaoseverlasting]

The pieces don't have to be together. We just have to divide the volume in such a was so as to get two equal parts right? So, you divide the slice symmetrically with one cut. The remaining part is also a rectangle, so you can cut that in several ways to get equal volumes. One of one part and another of the other and you have your equal volumes. What do you think?

 

[jim.nastics]

I had asked some other questions during the process of trying to answer this.

All angles are right angles. The planes of the sides of the missing rectangle are parallel to the planes of the sides of the cake. The cuts to make are vertical, but are not necessarily parallel to the sides of the cake. The cuts are straight lines.

No measuring and calculating, the solution must be general so it works for all possible rectangular shapes and locations.

 

[davee123]

No measuring and calculating

This makes me think that there is some key piece of information missing. If there's no measuring, that means cuts MUST be made from and to pre-determined points. This leaves cut starting points restricted to one of 8 places on the cake (effectively *2* places, since this is the general case), those being the corners of the cake and the corners of the rectangular piece that's been cut out. Ending places could be other corners, or perhaps we're granting the cutter enough skill to cut parallel to one of the cake's sides. Maybe we could even grant that the 2nd cut could be parallel and/or perpendicular to the 1st cut.

But still, this would appear to leave us with very few possibilities (relatively speaking).

Now... I have to ask-- the way you've phrased the question, it sounds like someone might have taken a cut arbitrarily out of the *CENTER* of the cake, surrounded on all 4 sides by cake. But you also give hints in your phrasing that the slice taken out *is* on an edge, just not on a corner. Could you clarify?

DaveE

 

[jim.nastics]

Micheal,

You just gave me an idea for a third wrong solution.

Dave,

I think that you are on the right track, somehow.

I do think that the only points that matter are the eight corners.

The missing rectangle is surrounded by cake. There may not be much cake, but it is a finite amount.

Since I see no reason to disallow your suggestion, I will grant that parallel and perpendicular cuts are within the skill of the knife wielder, if they are based on an existing edge and/or an existing set of points.

My problem with all of the two-cut solutions I considered is having them fail when I "moved" the missing piece around, or made it very wide or skinny relative to the cake, or very large or small.

 

[out of whack]

How is this:

A line that passes through the center of a rectangle divides it into two equal halves. We want to divide both the cake AND the empty area into two equal halves, so cut along a line that passes through the center of both rectangles to end up with equal volumes of cake. It's only one line, but it makes two cuts if you count each side of the missing rectangle as a separate cut.

 

[davee123]

A line that passes through the center of a rectangle

I think the solution works, but wouldn't that entail measuring?

DaveE

 

[out of whack]

I think the solution works, but wouldn't that entail measuring?

DaveE

Not with a measuring tape or ruler. Just aligning with an X on a rectangle should be allowed. You do need to define a specific direction for the cut, and you cannot do so without using any reference point at all.

 

[jim.nastics]

Out of whack,

It counts as one cut. Other than that, it is good a good answer. Thinking about the points you are using (centers of areas), they can be derived from the existing points using geometry. I don't think using geometry is "measuring".

Here is another wrong solution I based on Michaels thought...:


Identify two intersecting lines, one that divides the cake diagonally, and one that divides the missing rectangle diagonally.
Cut on these two intersecting lines.
There are now four pieces of cake.
(I think) the sums of the pairs of opposing pieces have equal volume.


While this is two cuts, the problem is that there are now two equal servings. the problem calls for two equal pieces.

 

[jim.nastics]

And I think I have the solution, need to think some more to decide if it is true for all cases...

 

[davee123]

Not with a measuring tape or ruler. Just aligning with an X on a rectangle should be allowed.

Oh, ok-- I figured that was expressly forbidden. I was assuming that "measuring" could only be done by making a cut-- hence, if you wanted to (say) find the midpoint of the cake, you could do so by making two cuts along the diagonals of the cake. Which of course reveals your midpoint, but wastes your two cuts.

Certainly if you allow that you can make marks on the cake by way of tracing lines and whatnot, I think that your solution works very well. In fact, it will work as a solution no matter WHAT the angle of the slice is, even if it's at a rakish angle in the middle or towards the edge of the cake. And... it only uses *one* cut.

Hm. Makes me think that there really is a different solution that necessitates two cuts, but does not require making marks or measuring of any kind-- just requires the cutter to be able to accurately cut in a straight line.

DaveE

 

[davee123]

Identify two intersecting lines, one that divides the cake diagonally, and one that divides the missing rectangle diagonally.
Cut on these two intersecting lines.
There are now four pieces of cake.
(I think) the sums of the pairs of opposing pieces have equal volume.

Hm, I don't think that one quite works-- try this with an inordinately wide but skinny slice, cut out near one side. IE, something like this:

+------------------------------------------+
|                                          |
|                                     +--+ |
|                                     |  | |
|                                     |  | |
|                                     |  | |
|                                     |  | |
|                                     |  | |
|                                     |  | |
|                                     |  | |
|                                     |  | |
|                                     |  | |
|                                     |  | |
|                                     +--+ |
|                                          |
+------------------------------------------+

DaveE

 

[jim.nastics]

Dave,

You are correct, my third solution does not work.

My fourth idea did not work either.

 

[Jimmy Snyder]

If there are two such holes in the cake, you can cut the cake into two pieces of equal volume with a single cut and you do not need to measure anything to determine how the cut is to be made. It doesn't matter whether the holes touch the edge of the cake or whether the holes intersect each other or whether the edges of the holes are parallel to the edges of the cake. The only assumption is that the cake and the holes are right rectangular prisms. The solution is hidden:

Determine the three points which are the centers of the cake, and of each of the two holes. This is done by finding the intersection points of the diagonals. Given three points, there is a plane that contains all three. Cut the cake along this plane.

eom

 

[Wizardsblade]

Hehe This one is fun. I think this is the answer (hiden text):

The first solution was really close. ok first of all you need 2 cuts to make 2 equal volumes. So each cut will only make a line in the cake. Put the cuts together to make a strait line through the cake of 2 equal volumes. The cake is not cut it horizontally, but vertically (through the side of the cake.) This will leave you with an identical top and bottom half ;).

 

[davee123]

The first solution was really close. ok first of all you need 2 cuts to make 2 equal volumes. So each cut will only make a line in the cake. Put the cuts together to make a strait line through the cake of 2 equal volumes. The cake is not cut it horizontally, but vertically (through the side of the cake.) This will leave you with an identical top and bottom half ;).

How is this different from the first "wrong" solution that jim.nastics posted?

DaveE

 

[Wizardsblade]

Basicly you have to use 2 cuts, "Using two separate cuts, divide the cake into two equal (volume) pieces". The first wrong solution states only 1 cut. I also must admit I missed (did not read) the first 2 wrong answers before I posted.

 

[DaveC426913]

Is it safe to make the following deduction?

Two cuts MUST be made.
The cuts must result in only TWO pieces of cake.

The only possible way to meet both these criteria is that the cuts, while they may START at an edge of the cake, they cannot END at an edge. This leaves only two possible end-points for each cut. They must end either in the hole or each other (or a combo of the two).

Hm. Maybe the opposite is more succinct: No cut can begin AND end on a cake edge.

 

[DaveC426913]

There is an ambiguity in the question IMO.

When you have the cake in two equal volume pieces, is that NO MORE THAN two pieces? You can't recombine smaller pieces to make one larger piece?

 

[Wizardsblade]

Is it safe to make the following deduction?

Two cuts MUST be made.
The cuts must result in only TWO pieces of cake.

The only possible way to meet both these criteria is that the cuts, while they may START at an edge of the cake, they cannot END at an edge. This leaves only two possible end-points for each cut. They must end either in the hole or each other (or a combo of the two).

Hm. Maybe the opposite is more succinct: No cut can begin AND end on a cake edge.

The cuts can both start and end on an edge. Take a knife put the point in the middle of a Cake now press down. You should have a line in the cake. Change your angle (probably want to change by 180 degrees) put the point in the middle again and press down. You have cut the cake in half with 2 cuts both started on top and went through to the bottom.

 

[Jimmy Snyder]

Change your angle (probably want to change by 180 degrees) put the point in the middle again and press down.

I like this solution. Since the cake can easily be cut into two pieces of equal volume with a single cut, all that remains is to split that single cut into two cuts.

 

[DaveC426913]

The cuts can both start and end on an edge. Take a knife put the point in the middle of a Cake now press down. You should have a line in the cake. Change your angle (probably want to change by 180 degrees) put the point in the middle again and press down. You have cut the cake in half with 2 cuts both started on top and went through to the bottom.

You didn't read very carefully.

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.

 

[davee123]

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:

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]

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...

then your amendment is unwarranted.


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

 

[AngeloG]

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.

 

[light_bulb]

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

doesn't end on an edge ;)

 

[davee123]

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...

then your amendment is unwarranted.

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]

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" 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.

 

[DaveC426913]

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.

 

[davee123]

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

 

[deanswift]

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]

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.