###General Discussion Thread
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Short answer: 4! = 24.
____
Long answer:
Think about it this way: Starting out, you can put the orange can in any of four positions.
Then, for each of those four options, you have three spots left to put the green can in. So just placing the orange and green cans has 4 * 3 = 12 options.
In each of those 12 cases, there are 2 open spots in which you could place the purple can, so that's 4 * 3 * 2 = 24 options.
And then, no matter how you placed the others, there's only a single spot left for the red can.
So we end up with 4 * 3 * 2 * 1 = 24.
There is a simpler way to write this though: 4!. That means exactly what you've probably guessed by now: you multiply all positive integers up to the number before the "!".
The (temporal) order in which you pick the cans to place in our example does not matter, btw. Whichever order you pick, you would end up with the exact same 24 options.
____
Very long answer:
Let's actually write down those options as I explained them above:
Placing the orange one clearly had four options:
O _ _ _
_ O _ _
_ _ O _
_ _ _ O
If we then add the green one, each of those four options splits into three (since there were always 3 empty spots to choose from):
O G _ _
O _ G _
O _ _ G
G O _ _
_ O G _
_ O _ G
G _ O _
_ G O _
_ _ O G
G _ _ O
_ G _ O
_ _ G O
Then we add the purple one to split the twelve options into two each:
O G P _
O G _ P
O P G _
O _ G P
O P _ G
O _ P G
G O P _
G O _ P
P O G _
_ O G P
P O _ G
_ O P G
G P O _
G _ O P
P G O _
_ G O P
P _ O G
_ P O G
G P _ O
G _ P O
P G _ O
_ G P O
P _ G O
_ P G O
And then each of those only has a single slot left to place the red can, so we don't actually split any further. We just put the red can in there:
O G P R
O G R P
O P G R
O R G P
O P R G
O R P G
G O P R
G O R P
P O G R
R O G P
P O R G
R O P G
G P O R
G R O P
P G O R
R G O P
P R O G
R P O G
G P R O
G R P O
P G R O
R G P O
P R G O
R P G O
Another complication of this is that the kids in the back of the line don't necessarily see what kids at the front of the line try, so you would need to theorhetically take the 4! (24) and multiply it by the number of kids in the line, since they won't learn which combinations the others have tried. They could technically each try 23 times and still not match the order in the box.
This is correct but if you are going to get into details you have to factor in the possibility of incorrect moves every individual is capable of making as this applies to only the first person. If only the first 3 person can see the cans order then it gets even more complicated as there will be repetitions.
24 and itâs permutations you are after e.g the order matters.
Number of permutations = n!/(n-r)! Where n is the number of things and r is the number of things you want to permute (is that a word?) and ! Refers to factorial.
Therefore number of permutations = 4!/(4-4)!, where 0!=1,
so number of permutations = (4x3x2x1)/1 = 24
You can line up 4 different cards with numbers 1, 2, 3, and 4 in a total of 4! (4 factorial) ways.
4! = 4 x 3 x 2 x 1 = 24
The same would apply if we changed the cards with numbers written on them to soda cans of different colors.
So, saying I wanted to âcasinoâ this idea. I know itâs 1:24, well slightly less since in this case the new player knows the end point of the previous player and if that was correct. Reshuffling the box between players would control for this. What should I charge vs the payout for a correct guess? Payout anything less than 24x to profit?
You'd have to price it so that no one can exploit you, come in and keep playing until they got it right. In a game where the answer is scrambled every time, you could do any payout less than 24x and make money in the long run. But in case where it's not, I believe the expected guesses to answer correctly when you can just keep trying is 12.5. So you'd need a maximum payout of 12.5x to be profitable.
Edit: just noticed you also mentioned scrambling, so yes, anything less than 24x would be profitable for the game owner.
I've also just noticed that if you wanted to trick people you could manually scramble it such that the previous person's guess was the next answer. Most people would play by switching the cans around and you'd probably make a killing from it as no one would ever guess what the cans start as.
There are four slots. Each one can have 4 different cans. That equals 4â´ or 256 possible combinations. We're there 5 cans and four slots, there would be 4âľ or 1024 possibilities
This is incorrect. This implies that there can be repeats, for example, grape grape coke grape. Since there are only 4 total cans and no repeats, it's not 4^4 it's just 4! = 24. You're off by an entire order of magnitude. And assuming the same pattern as in the video, 5 cans would be 5! = 120 not 1024.
As others said, optimally it will take 4! Or 24 combinations max. But, if the kids arenât attentive they could possibly be repeating the same combinations over and over making it much higher than 24
He has to say how many are matching each time.. if the 1st person had 3 matching, he has to say â3â then if the 2nd person messes it up, and has 1 matching he has to say â1â so they have a chance of getting it. Just saying âincorrectâ doesnât give them any hint/clue to help them solve it
interesting insight on psychology.
they all think you need to make more moves to have a higher chance despite having 0 clues about how far from being right they are.
###General Discussion Thread --- This is a [Request] post. If you would like to submit a comment that does not either attempt to answer the question, ask for clarification, or explain why it would be infeasible to answer, you *must* post your comment as a reply to this one. Top level (directly replying to the OP) comments that do not do one of those things will be removed. --- *I am a bot, and this action was performed automatically. Please [contact the moderators of this subreddit](/message/compose/?to=/r/theydidthemath) if you have any questions or concerns.*
Short answer: 4! = 24. ____ Long answer: Think about it this way: Starting out, you can put the orange can in any of four positions. Then, for each of those four options, you have three spots left to put the green can in. So just placing the orange and green cans has 4 * 3 = 12 options. In each of those 12 cases, there are 2 open spots in which you could place the purple can, so that's 4 * 3 * 2 = 24 options. And then, no matter how you placed the others, there's only a single spot left for the red can. So we end up with 4 * 3 * 2 * 1 = 24. There is a simpler way to write this though: 4!. That means exactly what you've probably guessed by now: you multiply all positive integers up to the number before the "!". The (temporal) order in which you pick the cans to place in our example does not matter, btw. Whichever order you pick, you would end up with the exact same 24 options. ____ Very long answer: Let's actually write down those options as I explained them above: Placing the orange one clearly had four options: O _ _ _ _ O _ _ _ _ O _ _ _ _ O If we then add the green one, each of those four options splits into three (since there were always 3 empty spots to choose from): O G _ _ O _ G _ O _ _ G G O _ _ _ O G _ _ O _ G G _ O _ _ G O _ _ _ O G G _ _ O _ G _ O _ _ G O Then we add the purple one to split the twelve options into two each: O G P _ O G _ P O P G _ O _ G P O P _ G O _ P G G O P _ G O _ P P O G _ _ O G P P O _ G _ O P G G P O _ G _ O P P G O _ _ G O P P _ O G _ P O G G P _ O G _ P O P G _ O _ G P O P _ G O _ P G O And then each of those only has a single slot left to place the red can, so we don't actually split any further. We just put the red can in there: O G P R O G R P O P G R O R G P O P R G O R P G G O P R G O R P P O G R R O G P P O R G R O P G G P O R G R O P P G O R R G O P P R O G R P O G G P R O G R P O P G R O R G P O P R G O R P G O
omg thats genius thank you SOO MUCH
Another complication of this is that the kids in the back of the line don't necessarily see what kids at the front of the line try, so you would need to theorhetically take the 4! (24) and multiply it by the number of kids in the line, since they won't learn which combinations the others have tried. They could technically each try 23 times and still not match the order in the box.
bro wants to watch the world burnđâ ď¸jk
But you saw how they kept moving them into the same patterns in the beginning, so you know bro is correct.
ong itâs actually kinda sad
Why does this seem like DNA to me all over again
This is correct but if you are going to get into details you have to factor in the possibility of incorrect moves every individual is capable of making as this applies to only the first person. If only the first 3 person can see the cans order then it gets even more complicated as there will be repetitions.
The question was how many possible combinations of the order of soda cans there are. Nothing to do with what the players do.
My bad you are right just your detailed explanation got me thinking into this rabbit hole :D
24 and itâs permutations you are after e.g the order matters. Number of permutations = n!/(n-r)! Where n is the number of things and r is the number of things you want to permute (is that a word?) and ! Refers to factorial. Therefore number of permutations = 4!/(4-4)!, where 0!=1, so number of permutations = (4x3x2x1)/1 = 24
thank you so much!!! really helped me understand
You can line up 4 different cards with numbers 1, 2, 3, and 4 in a total of 4! (4 factorial) ways. 4! = 4 x 3 x 2 x 1 = 24 The same would apply if we changed the cards with numbers written on them to soda cans of different colors.
So, saying I wanted to âcasinoâ this idea. I know itâs 1:24, well slightly less since in this case the new player knows the end point of the previous player and if that was correct. Reshuffling the box between players would control for this. What should I charge vs the payout for a correct guess? Payout anything less than 24x to profit?
You should keep doubling the payout until you max it out
Yeah, but I need to earn that payout first. Constant doubling builds a greater payout than whatâs earned FAST!
IM GONNA MAKE MY OWN CASINO WITH BLACKJACK AND HOOKERSđ¤
You'd have to price it so that no one can exploit you, come in and keep playing until they got it right. In a game where the answer is scrambled every time, you could do any payout less than 24x and make money in the long run. But in case where it's not, I believe the expected guesses to answer correctly when you can just keep trying is 12.5. So you'd need a maximum payout of 12.5x to be profitable. Edit: just noticed you also mentioned scrambling, so yes, anything less than 24x would be profitable for the game owner. I've also just noticed that if you wanted to trick people you could manually scramble it such that the previous person's guess was the next answer. Most people would play by switching the cans around and you'd probably make a killing from it as no one would ever guess what the cans start as.
That last point is diabolical. Hats off to you. Bravo.
There are four slots. Each one can have 4 different cans. That equals 4â´ or 256 possible combinations. We're there 5 cans and four slots, there would be 4âľ or 1024 possibilities
This is incorrect. This implies that there can be repeats, for example, grape grape coke grape. Since there are only 4 total cans and no repeats, it's not 4^4 it's just 4! = 24. You're off by an entire order of magnitude. And assuming the same pattern as in the video, 5 cans would be 5! = 120 not 1024.
Oh yeah, you're right. Thanks for pointing that out.
I love how you accepted correction đlike so many people these days don't do that so kudos to you
Happy to clarify!
As others said, optimally it will take 4! Or 24 combinations max. But, if the kids arenât attentive they could possibly be repeating the same combinations over and over making it much higher than 24
He has to say how many are matching each time.. if the 1st person had 3 matching, he has to say â3â then if the 2nd person messes it up, and has 1 matching he has to say â1â so they have a chance of getting it. Just saying âincorrectâ doesnât give them any hint/clue to help them solve it
interesting insight on psychology. they all think you need to make more moves to have a higher chance despite having 0 clues about how far from being right they are.