# Infinitely long loaf of Bread [closed]

You are visiting your old friend Mike at Infinitely's Baking Shop. Just as you arrived, he was taking out a fresh, infinitely long loaf of bread. Both of its ends extend infinitely long in a straight line. The smell of the breads displayed on the infinitely long shelves feels so good.

But you know that Jesse, your roommate, only eats the crusts of the bread (the ends of the bread). If you ask for the freshly baked bread, you'll grab it by its center, and since it's infinite in both directions, no finite speeds can bring you to either of its ends, and you know you can't travel faster than light anyway.

If you grab one of the loaves of bread from the display, you can grab it by one of its ends and thus cut a single crust piece, but Jesse needs two, or he'll stay hungry.

You might consider buying two loaves of bread from the display to solve this dilemma, but you can afford only one loaf and can't obtain more money for today by any means. Nor can you obtain more loaves by any other means. Mike is also not very giving, to just give you a loaf without receiving the money equaling its total cost.

After standing there shortly, you came up with an idea. At the end of the day, you walked out with two crusts in one hand, and the rest of the infinitely long bread in your infinitely long bag carried in your other hand. How did you manage to do it?

The picture represents a rough estimate of how a loaf of bread would look like if it was finite in length. (Mike only sells infinite breads)

The intended solution, in my opinion, is quite nice. I'm not sure if there are any other tricky means to solve the problem, but go for it.

Edit: The accepted answer, in my opinion, is simpler and better than the intended solution, and it can lead to a "problem" regarding infinities.

Intended solution hint:

The "problem" could be represented as a disagreement between yourself and Mike, which can take place some moments after Mike already agreed to your proposal.

Added the "open-ended" tag because it is unknown how many ways the infinities of the loaves can be interpreted.

• Why does Jesse have to be so picky? Jun 21 '17 at 15:40
• @BeastlyGerbil It's an infinitely long bag, as it says in the story. Jun 21 '17 at 15:42
• Also how has Mike not gone out off business? You only need one loaf to feed you for the rest of your life :P Jun 21 '17 at 15:43
• The real question here is how did you manage to stuff an infinitely long piece of bread into an infinite bag in a finite amount of time? Jun 21 '17 at 16:42
• @Carl Witthoft Everyone knows the true value of the bread is all in the crust at the ends. The rest is just filler. Any finite portion has no value, unless it is the end crust, in which case it contains half the value (with the other half in the other end crust). Jun 21 '17 at 20:29

You can ask for the freshly-baked bread, grabbing it by its center and

asking that it be packed into the bag with the bread folded at the point you grabbed it, and with that point going into the bag first. The two ends would be within easy reach of the bag's opening, so you just cut them off, hold the bag in one hand and the crusts in the other.

• TBH, this is simpler and better (in my opinion) than my initial solution, and also, my initial solution can lead to a "problem" regarding infinities as I've just realized, so I've decided to accept this instead. Jun 21 '17 at 16:01
• I'm curious to know what the intended solution is. Jun 21 '17 at 16:09
• @greenturtle3141 I've added a small hint for now. Jun 21 '17 at 16:35
• Comments are not for extended discussion; this conversation has been moved to chat.
– Rubio
Jun 23 '17 at 10:41
• @Lawrence I actually seemed to have read over the part where it is actually referred to as the center. My that's a lot of bread. Wonder if it has its own gravitational field? Jun 26 '17 at 6:47

You are a physicist (or at least, you have heard a few physicists chatting at the water cooler).

Mike, your friend, is very obliging. Not obliging enough to chop off both ends of a loaf for you, but at least obliging enough to provide you with a sliced loaf.

You greet Mike and grab one end of a loaf from his display, immediately slicing off a crust. Then you get Mike to slice the remainder evenly. Mike deals with infinite loaves every day, so this is easy for him, even if it is, to put it mildly, somewhat difficult for you to do.

You then get Mike to place the slices into the infinite bag in this order: 1 slice, then 2 slices, then 3 and so on, each time incrementing the number of slices thrown into the bag.

This takes almost the whole day (Mike is working at a furious pace towards the end), which is why you don't get out of the shop until the end of the day.

When Mike is done, you discover to your surprise that your bag now contains -1/12 of a slice of bread! You point this out to Mike, and ask for a proper loaf in exchange for the bag and its contents. Mike objects, saying that you still have a piece in your hand. You counter by saying that it would be unhygenic to return that piece, but he can keep one crust from the new loaf.

Mike mutters something under his breath while you grab a new loaf, surreptitiously slicing off the crust as you do so. Mike is already at the other end, slicing off the far crust as agreed. He wonders what he'll do with a finite piece of bread in his shop. He isn't as helpful now, unceremoniously dumping the rest of the loaf into a new bag and bidding you good day. You walk out with the bag in one hand and Jesse's two crusts in the other, determined to make it up to Mike by buying him a bottomless mug of coffee from the cafe next door once you've saved up enough.

• Utilizing lateral-thinking tag, one can interpret the sum of integers as $-1/12$ indeed. I also like the story element in this one. Jun 21 '17 at 16:26
• @Vepir I hope you enjoyed reading this answer as much as I did writing it. :) Jun 21 '17 at 16:31
• That infinite sum doesn't actually work. You can't rearrange the elements of a divergent series. There might be some kind of function or special number or mapping that you can apply to the infinite sum that gives back -1/12, but it doesn't converge to -1/12 that way. Jun 23 '17 at 4:36
• If you can surreptitiously slice one crust off while Mike slices the other off, you could have just done this with the initial loaf. This contradicts the third paragraph of the puzzle. Jun 23 '17 at 13:41
• @Rawling You can't reach the other end. What you have is one crust from the first loaf, and one from the second. Mike doesn't ever give to your hand the crust from the far end on its own - either it goes into the bag with the rest of the slices, or Mike keeps it for himself. Jun 23 '17 at 13:51

Trick question!

An infinitely long loaf of bread would, by definition, not have more than one end because it keeps going forever. As soon as you put two ends on a line it becomes a segment and is no longer infinite. Therefore, the only option is to take the ends from two loaves.

• @Vepir, Sorry, but a sequence of infinitely many of something can not be "followed by" something else. that's not what "infinitely many" means. You seem to be thinking of "infinity" as a number that happens to be bigger than any other number. But, there is no such number. Infinity is not a number. "Infinity" means "goes on forever." It means "without end" Jun 22 '17 at 13:36
• That depends on if Mike offers fractal bread or not. Jun 22 '17 at 14:10
• @jameslarge It surely can, it simply depends how you define it. You are mistaking the infinity here with a concept behind diverging set of elements. The most common definition of infinity is something that does not end, however if you construct a sentence like the one I just did, or define breads like that, one can have an infinite object with ends. Define the bread: Take a normal bread with ends and cut it in half. Separate the ends a bit and fill the gap with more bread. Repeat infinitely many times. You now have two ends of bread separated with infinitely many bread. :) Jun 22 '17 at 17:59
• @james large You're wrong. There is an infinite amount of numbers between 0 and 1. That's an infinite sequence despite having a defined start and end. Just consider that 0 and 1 are both ends of the loaf of bread. It's the same concept as an infinitely long piece of string. Jun 23 '17 at 11:53
• @jameslarge While there are still good points to be made about how the infinite bread should be interpreted, you absolutely can discuss the idea of "a sequence of infinitely many things" followed by another. For example, if we give a new order the whole numbers by saying "0 is the biggest, and the positive numbers are ordered normally below that", then we do have the infinitely many positive numbers followed by 0: 1<2<3<...<0. This is not nonsense, but exactly what the ordinal called ω+1 represents. Jun 25 '17 at 15:52

I ask Mike to bake a loaf specially for me which is cut in half at the centre before going into the oven. When it is removed from the oven, I cut off the crusts in the middle for Jesse.

• That was the intended solution! Spoiler ahead even if he does not bake on specific orders, simply wait until he starts baking a new one and ask for the same favor since you'll be buying it. And note the argument comes that at the point when it is baked, it seems you now have two breads for a price of one (since both parts have ends and are infinite)! But as soon as you cut the crusts and hide them away, then the two parts make a single cut loaf again, fixing the things, as Justin Ohms used this analogy in his answer.(Unless the crust is worth more than the rest of infinitely long bread?) Jun 22 '17 at 7:40
• @Vepir; Re, "you now have two breads for a price of one." Really? But in your presentation of the puzzle you said, "you can afford only one loaf, and can't obtain ... any more loafs by any other ways." You also implied that Mike does not negotiate; "Mike is also not very giving..." So if the indended solution requires Mike to graciously bake you a special order---two loaves for the price of one---then the whole puzzle seems rather shabby. Jun 22 '17 at 18:44

This is , so I would:

- Cut off a crust from the end I can reach.
- Slice laterally and get some crust from the "top end" of the bread. You only need a slice the same size as the other piece of crust.

I tried to only buy these two pieces, and Mike told me I had to buy the entire infinite loaf. Joke's on him, never going to need to buy bread again!

• what are you going to do in a month when you have infinite stale moldy bread? Jun 21 '17 at 20:11
• Tim Couwelier actually mentions this idea in the comments of the post. I guess the crust isn't precisely defined in the story either. Jun 22 '17 at 7:48
• My question (trying to figure out if this would be an accepted answer) predates this answer, my detailing the reasoning behind the question however only came after this answer. I suggest this answer be judged as such: as Laurel's answer and not just piggybacking on my comment. Jun 22 '17 at 10:59
• @ Sconibulus, mold spreads at a finite rate, just cut off the moldy part and make crutons from the stale part. Jun 22 '17 at 11:29
• Water- yes. Dryness- no. It's the old "speed of dark" trick. Jun 22 '17 at 15:02

This is actually a very easy problem, once you take into account the rules of the question.

1. Take an infinitely long loaf of bread from the shelf and cut off the crust on the end closest to you. (1/2)
2. Put the infinitely long loaf of bread into your infinitely long shopping bag (which the puzzle states we have and are capable of putting an infinitely long loaf of bread into), placing it with the cut crust down.
3. The other end of the infinitely long loaf of bread is now sitting flush with the top of your infinitely long shopping bag, and you can easily cut it off. (2/2)

Voila. One purchased loaf of bread, with two finite slices with crust.

I tell Mike that

I'll buy these two loaves of bread but first I want him to cut off the two ends that I can reach. Once he hands me the two ends I tuck them away and tell him that I changed my mind and I'll just take the one infinitely long loaf that he has sitting there, the one that he has sitting there cut in half. I'll grab the two ends that have no crust put them together and now I'm holding the middle of one infinitely long loaf.

Mike might get upset but it is only one infinitely long loaf that happens to be cut in half.

• Playing with infinities :) Spoiler ahead: this can be used to resolve the problem occuring in the intended solution, in your favor. (But if one assumes the crust has all the value, and the infinite bread part is just a filler, to make sense of finite prices, then you just stole from Mike!) Jun 22 '17 at 7:21

As any good bakery, Mike should offer to slice any bread for his customers. So I will ask him to cut the bread into infinite slices, handing me the first and the last slice in an extra bag. But I don't have infinite time to wait until he is finished. So how is he supposed to do infinite cuts in finite time?

I ask him to take one minute to cut off the first slice (and put it in the extra bag), 1/2 minute to cut off the second, 1/4 minute to cut off the third, 1/8 minute to cut off the fourth and so on. After two minutes, he will have a pile of infinite slices with the last slice on top, which he can then take off and also put in the extra bag.

If you insist that the "can only handle the bread from the center" rule also applies to Mike, he can:

start by cutting it in half, and then perform above algorithm on both halves one after another, handing me the last slice of each.

This method has the nice side-effect that I now have two bags of infinite slices of bread, so I can serve the second for breakfast in my Hilbert hotel.

My answer involves a bit of back-and-forth: first, purchase a freshly baked loaf and

Cut it in to two pieces, each of which are infinitely long.

Then you just have to trade

the two loaves for two from the shelf.

This is a fair trade since

Each of the two "half" loaves is infinite in extent, and therefore equal in value to a "whole" loaf.

• These two loaves are missing a crust each, at the cutting point. Wouldn't that make them tiny bit less, a cut of infinity? They are distinguishable from the loaves on the shelves. Jun 22 '17 at 7:29
• @Vepir That's not the problem. Infinity minus 1 is still infinity. But the true reason that the two halves are not equivalent is because they each have only one end-crust, which makes them half as valuable for anyone else who only likes the ends. Jun 22 '17 at 18:42

There is little practical difference between the crust on the end of the bread and the crust on the side. Therefore you can cut a six inch chunk of loaf off and then slice down the sides to take two "crusts" off the side. This has the wonderful advantage that Jesse can have more than two pieces of bread!

I would have posted the same answer as @PeterTaylor (+1) but as he beat me too it...

Cut off one end, take a single almost infinite Scooby-Doo-like bite out of the middle, then Cut off the other end. Although eating an almost infinite loaf of bread like that would test even the largest cartoon bellies...