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Are you smart, prove it? (:
So pretty much I work at Little Caesars as head supervisor, and my and one of my co-workers started a weird conversation and he stated that little caesars requires no knowledge of math, I'm not going to agree or disagree, but I decided to make a math problem revolving around what we do at Little Caesars, now I sort of would like a solution.
Introduction:
This is an INSANELY difficult math problem (possibly the most tedious math problem on the internet to solve), so only attempt if you're able to wrap your head around many ideas at once, and able to keep up with very many variables at once. Good luck (;
To understand the problem, first you have the understand the equipment being used here. You grab a tray of 12 dough balls from the cooler, you place these 12 dough balls in the flour. Now what you do is one at a time you make these dough balls into 2.5cm pancake type shapes, and you stack the on top of each other. There is space for two stacks to fit, so you can stack them 6 and 6, or 11 and 1 etc. Now you have to run them through a machine to flatten them, this makes them the ideal size, of about 0.8cm thick, the machine will make them 3x thinner as they were originally. After that they are put into pans, a 0.8cm "pizza sheetout" fits perfectly, it is your goal of what you're trying to achieve, this however being impossible, we can manipulate the goal as to... "how thick can I make the thinnest pizza?"
Problem:
Now the problem is, how many pizzas should you put in each stack to so you get best quality sheetouts. (As many as close to 0.8cm as possible). Should you rearrange the two stacks, ie. if you have 10 in one, 2 in the other, move 9 from one to the other stack so you can access the bottom ones? If so, how and when would you rearrange them.
Tell me exactly how I need to stack them, the order I need to do everything in, that the thinnest pizza made is as thick as possible.
Now background information that must be considered when doing this question.
1) Each of the 12 dough balls is made into the same thickness.
2) The thickest a sheetout can be made when running it through the machine is 2.5cm thick, any thicker wont fit.
3) The pancake shaped 2.5cm dough pieces can only be stacked in two places ontop of each other.
4) All dough balls need to be made into the pancake shaped 2.5cm pieces before you can begin running them through the machine.
Now where it gets really confusing... These are the properties of the dough balls. Understand that by putting weight on the dough balls you are squishing them, thereby, reducing their thickness.
1) If you put a pancake shaped dough ball onto another pancake shape doughball, the rate at which the dough ball will compress is 1/3 of it's original size every 60 seconds.
2) Now if there are two dough balls on top of a dough ball, the dough ball will be compressed at 1/3 of it's original size + 1/3 of the second dough ball, so the rate at which the the dough ball would get compressed at if there were 3 on top of it (4 dough ball in total) would be 1/3 + 1/9 + 1/27... My explanation wasn't the best, but I hope the example clears it up.
3) Now of course if that were true, the dough ball would approach a limit of zero, because that isn't true, what actually happens is at time = 0 the size decreases at 1/3 per 60 seconds. When the thickness is half of that of the original, the rate of decrease is 1/9 per 60 seconds if there's one dough ball above it, same principle applies for any extra dough balls above it, 1/81, 1/729 etc... This relationship continues, when the size is a quarter of the original, the size will now decrease at 1/81 per 60 seconds if there is one dough ball on top of it.
Now lets take a breather... Those are the mechanics of dough balls, now we're on to how long each task takes...
1) Taking a dough ball and making it into 2.5cm thick takes 10 seconds.
2) Running a dough ball through the machine takes 3 seconds, HOWEVER you are allowed to take two dough balls at once (one in each hand), but they have to be ran through the machine individually, so two pancake dough pieces would still take 6 second to flatten out.
3) You can decide to put the flattened pizza dough aside, or put it in the pan, however, if you put it aside (there is only one aside pile), they will begin to stick after 90 seconds of being together, -10 seconds for each extra one stacked... 2 flat pizza dough is 90 seconds, 3 flat pizza dough is 80 seconds, and so forth.
4) Placing pizza dough in the aside pile takes 1.5 seconds opposed to just putting the pizza dough in the pan, so consider whether it's worthwhile to use that precious time. Then you have to spend 5 more seconds putting it in the pan.
5) Putting dough into the pan takes 5 seconds.
Now remember when I said you can move dough from stack to stack? Well the process of moving the dough itself takes so time because it needs to be done carefully. Moving the dough (any sized stack) takes 4 seconds of time.
Now if you are able to solve this, props to you, if you answer with detailed proof, I'll send you a $50 futureshop giftcard, I don't really expect anyone to try, but yeah (:
Some parts may be a little bit confusing, so if you need me to reiterate a part or two, feel free to comment.
 
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Question1: The final goal is to put the pizza doughs in pans - and it appears putting it in a pan is quicker than putting it aside, so why would you need to put it aside and not just in a pan?
>> After that they are put into pans, a 0.8cm "pizza sheetout" fits perfectly, it is your goal of what you're trying to achieve.
>> 4) Placing pizza dough in the aside pile takes an extra 1.5 seconds opposed to just putting the pizza dough in the pan, so consider whether it's worthwhile to use that precious time.
5) Putting dough into the pan takes 5 seconds.
Question2: Why not use that "aside" place as a third 2.5cm dough pile and reduce the total flattening? (:
(eg: you put 4+4 at the two piles and 4 more at the "aside" place)
Question3: Do you have to finish all the initial dough ball flattening (into 2.5cm) before you can use the machine? Why not just make the pizza doughs one by one, before even flattening all balls? ((:
Question4: The machine makes 0.8cm out of 2.5 - then does it matter how much flattened the dough has gone between 2.5 and 0.8 in the meantime? Is the output the same? Then what is there to optimize??? :D
Remark: My question 2 and 3 are more troll / trick-playing ones, but the 1 and 4 are quite serious.
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Really slow day at Little Caesars eh?
This isn't a very good math problem. 
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On June 06 2012 15:09 figq wrote:
Question1: The final goal is to put the pizza doughs in pans - and it appears putting it in a pan is quicker than putting it aside, so why would you need to put it aside and not just in a pan?
>> After that they are put into pans, a 0.8cm "pizza sheetout" fits perfectly, it is your goal of what you're trying to achieve.
>> 4) Placing pizza dough in the aside pile takes an extra 1.5 seconds opposed to just putting the pizza dough in the pan, so consider whether it's worthwhile to use that precious time.
5) Putting dough into the pan takes 5 seconds.
Question2: Why not use that "aside" place as a third 2.5cm dough pile and reduce the total flattening? (:
Question 1: Sorry, I meant to write putting it aside takes 1.5 seconds, but then when you want to put it in the pan it takes 5 seconds, so putting it aside allows you to put other dough through the machine to prevent it from getting squished.
Question 2: There isn't very much space there, gotta have thin dough for it to fit (;
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On June 06 2012 15:12 Sinensis wrote:Really slow day at Little Caesars eh? This isn't a very good math problem.
Surprisingly not, I was actually just doing sheetouts, and I'm like hmm... I gotta work faster so my dough doesn't squish so much, then I started thinking about this haha.
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Thanks for the answers, I just added 2 more questions (above).
Also, what are we optimizing exactly? Time or thickness? If you insist on optimizing both... well, there may be many locally optimal solutions, because it seems with one by one (slowest) tactic you get the thickest pizza, and quicker tactics get you thinner pizza.
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On June 06 2012 15:16 FiWiFaKi wrote:Show nested quote +On June 06 2012 15:12 Sinensis wrote:Really slow day at Little Caesars eh? This isn't a very good math problem. Surprisingly not, I was actually just doing sheetouts, and I'm like hmm... I gotta work faster so my dough doesn't squish so much, then I started thinking about this haha.
When dough is cold it doesn't squish as much. Maybe you can leave it under refridgeration until it's closer to time to cook?
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On June 06 2012 15:21 figq wrote:
Thanks for the answers, I just added 2 more questions.
Also, what are we optimizing exactly? Time or thickness? If you insist on optimizing both... well, there may be many locally optimal solutions, because it seems with one by one (slowest) tactic you get the thickest pizza, and quicker tactics get you thinner pizza.
#3... It's just not practical, it eats away too much time to do them one by one, always having to switch tasks like that.
#4 I believe I did say that the machine makes it 3x thinner, So 2.5cm actually becomes 0.83333333, so I mean yeah, theoretically you could use a different size like 2.4cm, but it doesn't really make sense to do so.
And no problem (:
And you are going for achieving the thickest pizza... more specifically, the thinnest pizza is the thickest it can be.
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On June 06 2012 15:22 Sinensis wrote:Show nested quote +On June 06 2012 15:16 FiWiFaKi wrote:On June 06 2012 15:12 Sinensis wrote:Really slow day at Little Caesars eh? This isn't a very good math problem. Surprisingly not, I was actually just doing sheetouts, and I'm like hmm... I gotta work faster so my dough doesn't squish so much, then I started thinking about this haha. When dough is cold it doesn't squish as much. Maybe you can leave it under refridgeration until it's closer to time to cook?
We take it out of the cooler right before we sheet it out, and one tray takes maybe 5 minutes. And they do squish quite abit when you put 11 other pieces of dough on them, each weighing 12oz.
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On June 06 2012 15:23 FiWiFaKi wrote:
#4 I believe I did say that the machine makes it 3x thinner, So 2.5cm actually becomes 0.83333333, so I mean yeah, theoretically you could use a different size like 2.4cm, but it doesn't really make sense to do so. But I mean, what about when it's flattened to a thinner size - say from 2.5cm down to 1.8cm. Then the machine makes it 0.6cm or again just 0.8cm?
I imagine a machine that actually makes a fixed thickness out of a thicker piece (i.e. anything above 0.8 becomes exactly 0.8); which then makes irrelevant any flattening from 2.5 to 0.8 before the machine.
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On June 06 2012 15:34 figq wrote:Show nested quote +On June 06 2012 15:23 FiWiFaKi wrote:
#4 I believe I did say that the machine makes it 3x thinner, So 2.5cm actually becomes 0.83333333, so I mean yeah, theoretically you could use a different size like 2.4cm, but it doesn't really make sense to do so. But I mean, what about when it's flattened to a thinner size - say from 2.5cm down to 1.8cm. Then the machine makes it 0.6cm or again just 0.8cm? I imagine a machine that actually makes a fixed thickness out of a thicker piece (i.e. anything above 0.8 becomes exactly 0.8); which then makes irrelevant any flattening from 2.5 to 0.8 before the machine.
You have to think of it as it makes it one third, just like you said initially... 1.8cm into 0.6cm... I know it's hard to imagine, but think of the machine applying a certain force, and it just so happens to do that... ^^
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Your problem isn't well posed, there is no solution. You seem to be getting math confused with some weird slurry of telepathy and magic with number sprinkles
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On June 06 2012 15:38 FiWiFaKi wrote:Show nested quote +On June 06 2012 15:34 figq wrote:On June 06 2012 15:23 FiWiFaKi wrote:
#4 I believe I did say that the machine makes it 3x thinner, So 2.5cm actually becomes 0.83333333, so I mean yeah, theoretically you could use a different size like 2.4cm, but it doesn't really make sense to do so. But I mean, what about when it's flattened to a thinner size - say from 2.5cm down to 1.8cm. Then the machine makes it 0.6cm or again just 0.8cm? I imagine a machine that actually makes a fixed thickness out of a thicker piece (i.e. anything above 0.8 becomes exactly 0.8); which then makes irrelevant any flattening from 2.5 to 0.8 before the machine. You have to think of it as it makes it one third, just like you said initially... 1.8cm into 0.6cm... I know it's hard to imagine, but think of the machine applying a certain force, and it just so happens to do that... ^^ Okay then! But I don't believe you that the real machine does that, can't imagine it. I accept it as a theoretical problem, but I'm very surprised if the real machine actually works like that. It seems so much easier for the machine to output fixed thickness.
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On June 06 2012 15:42 figq wrote:Show nested quote +On June 06 2012 15:38 FiWiFaKi wrote:On June 06 2012 15:34 figq wrote:On June 06 2012 15:23 FiWiFaKi wrote:
#4 I believe I did say that the machine makes it 3x thinner, So 2.5cm actually becomes 0.83333333, so I mean yeah, theoretically you could use a different size like 2.4cm, but it doesn't really make sense to do so. But I mean, what about when it's flattened to a thinner size - say from 2.5cm down to 1.8cm. Then the machine makes it 0.6cm or again just 0.8cm? I imagine a machine that actually makes a fixed thickness out of a thicker piece (i.e. anything above 0.8 becomes exactly 0.8); which then makes irrelevant any flattening from 2.5 to 0.8 before the machine. You have to think of it as it makes it one third, just like you said initially... 1.8cm into 0.6cm... I know it's hard to imagine, but think of the machine applying a certain force, and it just so happens to do that... ^^ Okay then! But I don't believe you that the real machine does that, can't imagine it. I accept it as a theoretical problem, but I'm very surprised if the real machine actually works like that. It seems so much easier for the machine to output fixed thickness.
Omfgg, I wanted to create a math problem, if it was like that, it wouldn't work! Sheesh ;p ... Go with the flow.
On June 06 2012 15:39 n.DieJokes wrote:
Your problem isn't well posed, there is no solution. You seem to be getting math confused with some weird slurry of telepathy and magic with number sprinkles
There has to be an answer, and there is nothing that would make me think there isn't. Once process will be the fastest, it's all about finding a method that'll determine that. And no idea what you mean, this is a real life application of math principles combined with logical thinking.
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Alright, I think now the setting is finally fully clear to me (thanks for clarifications!), except for this part:
On June 06 2012 15:23 FiWiFaKi wrote:
#3... It's just not practical, it eats away too much time to do them one by one, always having to switch tasks like that. Does that mean all 12 dough balls are finished being flattened and put on the 2 piles before we start using the machine?
Can we switch around stacks, while still flattening the initial dough balls?
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On June 06 2012 15:39 n.DieJokes wrote:
Your problem isn't well posed, there is no solution. You seem to be getting math confused with some weird slurry of telepathy and magic with number sprinkles
I have to agree with n.die and your coworker.
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This is just one of those things where you have to be there to understand what the hell is going on. A youtube video is worth a book's worth of your descriptions.
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Here is my $5 GIMMIE PIZZZAAAA PIZZZAA PIZZZAAAAAAAAAAAAAAA
Edit: Originally put 1 too many pizza's...
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I'm going to assume that this can be brute force solved with enough variables and some lagrange multipliers.
But I'm not gonna do it - ________ -;
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On June 06 2012 15:58 figq wrote:Alright, I think now the setting is finally fully clear to me (thanks for clarifications!), except for this part: Show nested quote +On June 06 2012 15:23 FiWiFaKi wrote:
#3... It's just not practical, it eats away too much time to do them one by one, always having to switch tasks like that. Does that mean all 12 dough balls are finished being flattened and put on the 2 piles before we start using the machine? Can we switch around stacks, while still flattening the initial dough balls?
Yes exactly, and yes to the switching around too (:
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I don't think this makes sense.
It takes 10 seconds to make a dough ball. It takes 3 seconds to put through the machine. It takes 5 seconds to put in a pan.
It seems these times are fixed for each dough ball, and fiddling with stacking and putting stuff aside won't change that. So as someone else said, making one dough ball at a time is the optimal solution with the information given.
Either that or hiring one more person, so that one person flattens balls (a great job description), and the other person machines flattened balls (an even better job description) and puts them in the pan. Thus you cut the time taken to 10 seconds from 18.
(I think you have to include how long it takes for you to switch tasks, to actually have a problem here to solve...)
Something to note:
Original piece thickness is 2.5cm, and optimal piece thickness is 2.4cm. 60 seconds in stack with 1 on top means decrease of 1/3 size (0.83cm). So 10 seconds in stack with 1 on top means a decrease of 0.14cm.
Since it takes 10 seconds to make the dough piece, the bottom piece in the stack is already too thin when you add the 3rd piece in a stack of 3. This will obviously continue the more you add. So when you have a stack of 12, every single one apart from the top 1 or 2 will be below the proper size.
I bet you're not meant to stack the things at all... you're just lazy. :p
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On June 07 2012 01:58 netherh wrote:
I don't think this makes sense.
It takes 10 seconds to make a dough ball. It takes 3 seconds to put through the machine. It takes 5 seconds to put in a pan.
It seems these times are fixed for each dough ball, and fiddling with stacking and putting stuff aside won't change that. So as someone else said, making one dough ball at a time is the optimal solution with the information given.
Either that or hiring one more person, so that one person flattens balls (a great job description), and the other person machines flattened balls (an even better job description) and puts them in the pan. Thus you cut the time taken to 10 seconds from 18.
(I think you have to include how long it takes for you to switch tasks, to actually have a problem here to solve...)
Something to note:
Original piece thickness is 2.5cm, and optimal piece thickness is 2.4cm. 60 seconds in stack with 1 on top means decrease of 1/3 size (0.83cm). So 10 seconds in stack with 1 on top means a decrease of 0.14cm.
Since it takes 10 seconds to make the dough piece, the bottom piece in the stack is already too thin when you add the 3rd piece in a stack of 3. This will obviously continue the more you add. So when you have a stack of 12, every single one apart from the top 1 or 2 will be below the proper size.
I bet you're not meant to stack the things at all... you're just lazy. :p
It's a question, this is the way it HAS to be done for the problem, so please do so accordingly, it's not whether the situation is realistic. Thanks.
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EPT
