Excellent Math Riddle
15 May 2013, 8:57 pm
One thing to keep in mind is that if you use hats 1 to 1001 instead of 0 to 1000, the my examples above still work. The only difference is that the binary operations are a bit different.
For example, if you do the addition by subtracting 1001 when the sum is greater than 1001 (which seems to be what several here have referred to as modular arithmetic), then 1001 becomes your additive identity element instead of 0.
15 May 2013, 10:09 pm
eric76 wrote:
aequitas1 wrote:
I felt and still feel that there needs to be some new dimension to the communication or guesses in this problem but I'm running through other algorithms but I feel like 998 is unsolvable without the second wizard being able to remain silent.
Being able to stay silent doesn't help. The more hat numbers are not eliminated, the sooner that there will be ambiguities.
If the first wizard says 4 like in the example below, he forces wizard 5 to give one of the two values creating uncertainty. It creates uncertainty rather than giving the wizards more information. Silence on his part would allow the string of wizards guessing their own hat value continue on to the last one. Here's an example:
Wizard 1* [ 5, 1000] mod 1001: 5 + 1000 = 4 thus guesses 4.
------------------------
Wizard 2 [99, 5, 1000] Wizards from here to 5 can determine their hat value.
Wizard 3 [10, 5, 1000]
Wizard 4 [37, 5, 1000]
---------------------------
Wizard 5* [4, 5, 1000] Negation of 4 so 5 or 1000 as a guess.
---------------------------
Wizard 6 [769, 5, 0]
Wizard 7 [16, 5, 0]
Wizard 8 [236, 5, 0] This wizard, supposing wizard 5 guesses one of the two numbers, can assume either wizard 5 or 6 were unable to guess their numbers.
Wizard 9 [690, 5, 0]
Wizard 10 [800, 5, 0]
Wizard 11 [33, 5, 0]
Wizard 12 [12, 5, 0]
Wizard 13 [205, 5, 0]
Wizard 14 [113, 5, 0]
Wizard 15 [800, 5, 0]
16 May 2013, 12:23 am
aequitas1 wrote:
eric76 wrote:
aequitas1 wrote:
I felt and still feel that there needs to be some new dimension to the communication or guesses in this problem but I'm running through other algorithms but I feel like 998 is unsolvable without the second wizard being able to remain silent.
Being able to stay silent doesn't help. The more hat numbers are not eliminated, the sooner that there will be ambiguities.
If the first wizard says 4 like in the example below, he forces wizard 5 to give one of the two values creating uncertainty. It creates uncertainty rather than giving the wizards more information. Silence on his part would allow the string of wizards guessing their own hat value continue on to the last one. Here's an example:
Wizard 1* [ 5, 1000] mod 1001: 5 + 1000 = 4 thus guesses 4.
------------------------
Wizard 2 [99, 5, 1000] Wizards from here to 5 can determine their hat value.
Wizard 3 [10, 5, 1000]
Wizard 4 [37, 5, 1000]
---------------------------
Wizard 5* [4, 5, 1000] Negation of 4 so 5 or 1000 as a guess.
---------------------------
Wizard 6 [769, 5, 0]
Wizard 7 [16, 5, 0]
Wizard 8 [236, 5, 0] This wizard, supposing wizard 5 guesses one of the two numbers, can assume either wizard 5 or 6 were unable to guess their numbers.
Wizard 9 [690, 5, 0]
Wizard 10 [800, 5, 0]
Wizard 11 [33, 5, 0]
Wizard 12 [12, 5, 0]
Wizard 13 [205, 5, 0]
Wizard 14 [113, 5, 0]
Wizard 15 [800, 5, 0]
When you are considering only the first very few terms in a very long sequence, it can appear to work. Try it for the entire sequence for a number of different distributions of hats and see what happens. You will see that ambiguities will arise.
What you really need to do is figure out the worst possible distribution of hats and calculate how many wizards will be able to determine their hat number from the information given.
If you succeed at that, I think that you will find that the number of wizards who are able to determine their hat number from the information given (i.e. guesses don't count) to be 0, not 998. That's right. Under the worst possible distribution of hats, by using this method you probably cannot guarantee that any wizard will make it.
What would actually be really interesting would be to determine the number of wizards that can determine their hat number from the information given for the best of all possible hat distributions.
16 May 2013, 1:27 am
eric76 wrote:
aequitas1 wrote:
eric76 wrote:
aequitas1 wrote:
I felt and still feel that there needs to be some new dimension to the communication or guesses in this problem but I'm running through other algorithms but I feel like 998 is unsolvable without the second wizard being able to remain silent.
Being able to stay silent doesn't help. The more hat numbers are not eliminated, the sooner that there will be ambiguities.
If the first wizard says 4 like in the example below, he forces wizard 5 to give one of the two values creating uncertainty. It creates uncertainty rather than giving the wizards more information. Silence on his part would allow the string of wizards guessing their own hat value continue on to the last one. Here's an example:
Wizard 1* [ 5, 1000] mod 1001: 5 + 1000 = 4 thus guesses 4.
------------------------
Wizard 2 [99, 5, 1000] Wizards from here to 5 can determine their hat value.
Wizard 3 [10, 5, 1000]
Wizard 4 [37, 5, 1000]
---------------------------
Wizard 5* [4, 5, 1000] Negation of 4 so 5 or 1000 as a guess.
---------------------------
Wizard 6 [769, 5, 0]
Wizard 7 [16, 5, 0]
Wizard 8 [236, 5, 0] This wizard, supposing wizard 5 guesses one of the two numbers, can assume either wizard 5 or 6 were unable to guess their numbers.
Wizard 9 [690, 5, 0]
Wizard 10 [800, 5, 0]
Wizard 11 [33, 5, 0]
Wizard 12 [12, 5, 0]
Wizard 13 [205, 5, 0]
Wizard 14 [113, 5, 0]
Wizard 15 [800, 5, 0]
When you are considering only the first very few terms in a very long sequence, it can appear to work. Try it for the entire sequence for a number of different distributions of hats and see what happens. You will see that ambiguities will arise.
What you really need to do is figure out the worst possible distribution of hats and calculate how many wizards will be able to determine their hat number from the information given.
If you succeed at that, I think that you will find that the number of wizards who are able to determine their hat number from the information given (i.e. guesses don't count) to be 0, not 998. That's right. Under the worst possible distribution of hats, by using this method you probably cannot guarantee that any wizard will make it.
What would actually be really interesting would be to determine the number of wizards that can determine their hat number from the information given for the best of all possible hat distributions.
Ignore the cardinal assignments of each wizard I've given because it holds true under any distribution of hat values. Nonetheless, I'm trying to solve with certainty for 998 so the fact that this particular distribution has a probability of failure flies in the face of certainty (or that the probabilities for success vary by distribution for that matter.) Regardless, this logic holds absolutely true for all configurations of hat assignments (unless wizard 1000 is in the forced guess position which would successfully promote 998). Somberlain is very quick and discerned this problem days ago and I've been sitting here trying to disprove it by establishing an algorithm the wizards can use.
Your logic is conventional and it may seem like one wizard remaining silent would produce more unknowns or ambiguity on the subsequent wizards but it doesn't. It would seem like it from first glance though. I'll give you an example for your argument:
Suppose Wizard 1, the wizard tasked with giving up his guess to express the sum, guesses 4 to express for a sum of 1005. Say Wizard 801 has a hat value of 4 and thus must posit a guess of either 5 or 1000 because his guess was taken. He sees three missing numbers: 4, 5, 1000. He simply calculates the sum total of each permutation and the one equal to Wizard 1's expression of 4 (which either is 4 or 1005) is one of the two numbers originally missing. Even though he can deduce his own hat number from this he cannot guess it. Subsequent to his guessing either 5 or 1000: if Wizard 802 has a value of 769 and Wizard 942 has a hat value of 236 then Wizard 960 having a hat value of 236 (from my analysis as of right now, at least) cannot know whether or not 5 or 236 is the value on his hat number because he has no way of knowing which of the two wizards, either Wizard 801 or Wizard 802 was the wizard forced to release one of the two combinations equal to 4 (or 1005).
Edit: under the worst case scenario you can still promote at least 2 wizards with this strategy.
Edit 1: I thought about constructing an algorithm for each wizard that included stats analysis but I still want a more optimal strategy proving 998.
Last edited by aequitas1 on 16 May 2013, 1:41 am, edited 1 time in total.
16 May 2013, 1:40 am
Go back to this example:
Quote:
First, assume that the hats are numbered 0-1000 and that all additions are mod 1001.
Suppose that hat 999 is not being worn and that the first three wizards to guess have numbers 12, 10, and 0 in that order.
The first wizard sees that 999 and 12 are missing. So he adds them together to get 12+999 = 10 and says the number 10.
The second wizard doesn't see hats 10, 12, and 999 and concludes that his hat must be 10 but that it is already taken so he stays silent.
The third wizard doesn't see hats 0, 10, 12, and 999. The problem is that 0+10=10 and 12+999=10. What number is he going to call out?
Suppose that hat 999 is not being worn and that the first three wizards to guess have numbers 12, 10, and 0 in that order.
The first wizard sees that 999 and 12 are missing. So he adds them together to get 12+999 = 10 and says the number 10.
The second wizard doesn't see hats 10, 12, and 999 and concludes that his hat must be 10 but that it is already taken so he stays silent.
The third wizard doesn't see hats 0, 10, 12, and 999. The problem is that 0+10=10 and 12+999=10. What number is he going to call out?
Starting the index with hat 0 and using real modular arithmetic, if no wizard has hat 999 and the hat numbers of the first three wizards are 12, 10, and 0, what numbers do the first three wizards call?
Last edited by eric76 on 16 May 2013, 1:42 am, edited 1 time in total.
16 May 2013, 1:44 am
eric76 wrote:
Go back to this example:
Starting the index with hat 0 and using real modular arithmetic, if no wizard has hat 999 and the hat numbers of the first three wizards are 12, 10, and 0, what numbers do the first three wizards call?
Quote:
First, assume that the hats are numbered 0-1000 and that all additions are mod 1001.
Suppose that hat 999 is not being worn and that the first three wizards to guess have numbers 12, 10, and 0 in that order.
The first wizard sees that 999 and 12 are missing. So he adds them together to get 12+999 = 10 and says the number 10.
The second wizard doesn't see hats 10, 12, and 999 and concludes that his hat must be 10 but that it is already taken so he stays silent.
The third wizard doesn't see hats 0, 10, 12, and 999. The problem is that 0+10=10 and 12+999=10. What number is he going to call out?
Suppose that hat 999 is not being worn and that the first three wizards to guess have numbers 12, 10, and 0 in that order.
The first wizard sees that 999 and 12 are missing. So he adds them together to get 12+999 = 10 and says the number 10.
The second wizard doesn't see hats 10, 12, and 999 and concludes that his hat must be 10 but that it is already taken so he stays silent.
The third wizard doesn't see hats 0, 10, 12, and 999. The problem is that 0+10=10 and 12+999=10. What number is he going to call out?
Starting the index with hat 0 and using real modular arithmetic, if no wizard has hat 999 and the hat numbers of the first three wizards are 12, 10, and 0, what numbers do the first three wizards call?
While it is mathematically sound to construct a set beginning at 0, the only construction you can apply is cardinal assignments to the wizards but the riddle states that the values of hats is ranged 1-1001 so in this case 0 is a construct being applied and ultimately ignored because hat 0 is non-existent.
16 May 2013, 1:50 am
aequitas1 wrote:
eric76 wrote:
Go back to this example:
Starting the index with hat 0 and using real modular arithmetic, if no wizard has hat 999 and the hat numbers of the first three wizards are 12, 10, and 0, what numbers do the first three wizards call?
Quote:
First, assume that the hats are numbered 0-1000 and that all additions are mod 1001.
Suppose that hat 999 is not being worn and that the first three wizards to guess have numbers 12, 10, and 0 in that order.
The first wizard sees that 999 and 12 are missing. So he adds them together to get 12+999 = 10 and says the number 10.
The second wizard doesn't see hats 10, 12, and 999 and concludes that his hat must be 10 but that it is already taken so he stays silent.
The third wizard doesn't see hats 0, 10, 12, and 999. The problem is that 0+10=10 and 12+999=10. What number is he going to call out?
Suppose that hat 999 is not being worn and that the first three wizards to guess have numbers 12, 10, and 0 in that order.
The first wizard sees that 999 and 12 are missing. So he adds them together to get 12+999 = 10 and says the number 10.
The second wizard doesn't see hats 10, 12, and 999 and concludes that his hat must be 10 but that it is already taken so he stays silent.
The third wizard doesn't see hats 0, 10, 12, and 999. The problem is that 0+10=10 and 12+999=10. What number is he going to call out?
Starting the index with hat 0 and using real modular arithmetic, if no wizard has hat 999 and the hat numbers of the first three wizards are 12, 10, and 0, what numbers do the first three wizards call?
While it is mathematically sound to construct a set beginning at 0, the only construction you can apply is cardinal assignments to the wizards but the riddle states that the values of hats is ranged 1-1001 so in this case 0 is a construct being applied and ultimately ignored because hat 0 is non-existent.
The problems are equivalent. The difference is the location of the additive identity.
Okay. Let's use 1-1001.
If you are adding the two numbers and subtracting 1001 if the sum is greater than 1001, then it is similar to modular arithmetic but it is not modular arithmetic. What you have basically done is move the additive identity from 0 to 1001.
If that is your method, then assume the hat no wizard is wearing is 999 and that the first three wizards are wearing hats 12, 10, and 1001 in that order.
If you are not adding the numbers as described above, then please explain your addition in detail.
16 May 2013, 1:52 am
eric76 wrote:
I don't think you can guarantee anything better than 500.
That is what I'm chasing and I'm hoping for a eureka! moment. It really doesn't look provable though after mulling it over.
I knew upon encountering the riddle that a strategy considerate of odd/even would solve for 500 but it didn't interest me as much as the allure of a more optimal solution drew me in. Somberlain had it so right though and his strategy for solving 500 was so elegant.
I'm still wondering if there is some kind of algorithm that would work but from where I'm sitting it may be impossible without an extra dimension added to at least 2 of the guesses. Somberlain posted about modular arithmetic and the example used was a clock and it relates closely to this problem: how can you express whether it is am or pm on a grandfather clock? It is impossible. I'm hoping for a profound difference in this problem with the clock problem but I'm not seeing one.
16 May 2013, 2:00 am
eric76 wrote:
aequitas1 wrote:
eric76 wrote:
Go back to this example:
Starting the index with hat 0 and using real modular arithmetic, if no wizard has hat 999 and the hat numbers of the first three wizards are 12, 10, and 0, what numbers do the first three wizards call?
Quote:
First, assume that the hats are numbered 0-1000 and that all additions are mod 1001.
Suppose that hat 999 is not being worn and that the first three wizards to guess have numbers 12, 10, and 0 in that order.
The first wizard sees that 999 and 12 are missing. So he adds them together to get 12+999 = 10 and says the number 10.
The second wizard doesn't see hats 10, 12, and 999 and concludes that his hat must be 10 but that it is already taken so he stays silent.
The third wizard doesn't see hats 0, 10, 12, and 999. The problem is that 0+10=10 and 12+999=10. What number is he going to call out?
Suppose that hat 999 is not being worn and that the first three wizards to guess have numbers 12, 10, and 0 in that order.
The first wizard sees that 999 and 12 are missing. So he adds them together to get 12+999 = 10 and says the number 10.
The second wizard doesn't see hats 10, 12, and 999 and concludes that his hat must be 10 but that it is already taken so he stays silent.
The third wizard doesn't see hats 0, 10, 12, and 999. The problem is that 0+10=10 and 12+999=10. What number is he going to call out?
Starting the index with hat 0 and using real modular arithmetic, if no wizard has hat 999 and the hat numbers of the first three wizards are 12, 10, and 0, what numbers do the first three wizards call?
While it is mathematically sound to construct a set beginning at 0, the only construction you can apply is cardinal assignments to the wizards but the riddle states that the values of hats is ranged 1-1001 so in this case 0 is a construct being applied and ultimately ignored because hat 0 is non-existent.
The problems are equivalent. The difference is the location of the additive identity.
Okay. Let's use 1-1001.
If you are adding the two numbers and subtracting 1001 if the sum is greater than 1001, then it is similar to modular arithmetic but it is not modular arithmetic. What you have basically done is move the additive identity from 0 to 1001.
If that is your method, then assume the hat no wizard is wearing is 999 and that the first three wizards are wearing hats 12, 10, and 1001 in that order.
If you are not adding the numbers as described above, then please explain your addition in detail.
I agree but it still seems as though the output would be exactly the same if the hats were numbered 0-1000 or 1-1001. The same uncertainty would apply.
16 May 2013, 2:14 am
eric76 wrote:
Go back to this example:
Starting the index with hat 0 and using real modular arithmetic, if no wizard has hat 999 and the hat numbers of the first three wizards are 12, 10, and 0, what numbers do the first three wizards call?
Quote:
First, assume that the hats are numbered 0-1000 and that all additions are mod 1001.
Suppose that hat 999 is not being worn and that the first three wizards to guess have numbers 12, 10, and 0 in that order.
The first wizard sees that 999 and 12 are missing. So he adds them together to get 12+999 = 10 and says the number 10.
The second wizard doesn't see hats 10, 12, and 999 and concludes that his hat must be 10 but that it is already taken so he stays silent.
The third wizard doesn't see hats 0, 10, 12, and 999. The problem is that 0+10=10 and 12+999=10. What number is he going to call out?
Suppose that hat 999 is not being worn and that the first three wizards to guess have numbers 12, 10, and 0 in that order.
The first wizard sees that 999 and 12 are missing. So he adds them together to get 12+999 = 10 and says the number 10.
The second wizard doesn't see hats 10, 12, and 999 and concludes that his hat must be 10 but that it is already taken so he stays silent.
The third wizard doesn't see hats 0, 10, 12, and 999. The problem is that 0+10=10 and 12+999=10. What number is he going to call out?
Starting the index with hat 0 and using real modular arithmetic, if no wizard has hat 999 and the hat numbers of the first three wizards are 12, 10, and 0, what numbers do the first three wizards call?
Excuse my ignorance on this construction from 0. I never learned basic mathematics because I could never stay focused in class and could only learn when I was alone. I understand things like fourier analysis more than I understand why one would construct a set beginning with 0.
The internet is a beautiful thing. So much to learn!
Last edited by aequitas1 on 16 May 2013, 2:16 am, edited 1 time in total.
16 May 2013, 2:16 am
aequitas1 wrote:
eric76 wrote:
aequitas1 wrote:
eric76 wrote:
Go back to this example:
Starting the index with hat 0 and using real modular arithmetic, if no wizard has hat 999 and the hat numbers of the first three wizards are 12, 10, and 0, what numbers do the first three wizards call?
Quote:
First, assume that the hats are numbered 0-1000 and that all additions are mod 1001.
Suppose that hat 999 is not being worn and that the first three wizards to guess have numbers 12, 10, and 0 in that order.
The first wizard sees that 999 and 12 are missing. So he adds them together to get 12+999 = 10 and says the number 10.
The second wizard doesn't see hats 10, 12, and 999 and concludes that his hat must be 10 but that it is already taken so he stays silent.
The third wizard doesn't see hats 0, 10, 12, and 999. The problem is that 0+10=10 and 12+999=10. What number is he going to call out?
Suppose that hat 999 is not being worn and that the first three wizards to guess have numbers 12, 10, and 0 in that order.
The first wizard sees that 999 and 12 are missing. So he adds them together to get 12+999 = 10 and says the number 10.
The second wizard doesn't see hats 10, 12, and 999 and concludes that his hat must be 10 but that it is already taken so he stays silent.
The third wizard doesn't see hats 0, 10, 12, and 999. The problem is that 0+10=10 and 12+999=10. What number is he going to call out?
Starting the index with hat 0 and using real modular arithmetic, if no wizard has hat 999 and the hat numbers of the first three wizards are 12, 10, and 0, what numbers do the first three wizards call?
While it is mathematically sound to construct a set beginning at 0, the only construction you can apply is cardinal assignments to the wizards but the riddle states that the values of hats is ranged 1-1001 so in this case 0 is a construct being applied and ultimately ignored because hat 0 is non-existent.
The problems are equivalent. The difference is the location of the additive identity.
Okay. Let's use 1-1001.
If you are adding the two numbers and subtracting 1001 if the sum is greater than 1001, then it is similar to modular arithmetic but it is not modular arithmetic. What you have basically done is move the additive identity from 0 to 1001.
If that is your method, then assume the hat no wizard is wearing is 999 and that the first three wizards are wearing hats 12, 10, and 1001 in that order.
If you are not adding the numbers as described above, then please explain your addition in detail.
I agree but it still seems as though the output would be exactly the same if the hats were numbered 0-1000 or 1-1001. The same uncertainty would apply.
They can't be exactly the same because the output would include a hat 1001, but no hat 0. The output from the same methods should be equivalent. If you are doing it like I think you are doing it, then your hat 1001 and my hat 0 are the same hat.
If the hats are numbered 1-1001, then you are not adding them together using modular arithmetic because in modular arithmetic, 1001 mod 1001 = 0. That is, you have to have a hat 0 and there cannot be a hat 1001.
That's why it makes sense to use the numbers 0 to 1000 instead of 1 to 1001 -- because you can then use modular arithmetic which is well understood.
16 May 2013, 2:18 am
eric76 wrote:
aequitas1 wrote:
eric76 wrote:
aequitas1 wrote:
eric76 wrote:
Go back to this example:
Starting the index with hat 0 and using real modular arithmetic, if no wizard has hat 999 and the hat numbers of the first three wizards are 12, 10, and 0, what numbers do the first three wizards call?
Quote:
First, assume that the hats are numbered 0-1000 and that all additions are mod 1001.
Suppose that hat 999 is not being worn and that the first three wizards to guess have numbers 12, 10, and 0 in that order.
The first wizard sees that 999 and 12 are missing. So he adds them together to get 12+999 = 10 and says the number 10.
The second wizard doesn't see hats 10, 12, and 999 and concludes that his hat must be 10 but that it is already taken so he stays silent.
The third wizard doesn't see hats 0, 10, 12, and 999. The problem is that 0+10=10 and 12+999=10. What number is he going to call out?
Suppose that hat 999 is not being worn and that the first three wizards to guess have numbers 12, 10, and 0 in that order.
The first wizard sees that 999 and 12 are missing. So he adds them together to get 12+999 = 10 and says the number 10.
The second wizard doesn't see hats 10, 12, and 999 and concludes that his hat must be 10 but that it is already taken so he stays silent.
The third wizard doesn't see hats 0, 10, 12, and 999. The problem is that 0+10=10 and 12+999=10. What number is he going to call out?
Starting the index with hat 0 and using real modular arithmetic, if no wizard has hat 999 and the hat numbers of the first three wizards are 12, 10, and 0, what numbers do the first three wizards call?
While it is mathematically sound to construct a set beginning at 0, the only construction you can apply is cardinal assignments to the wizards but the riddle states that the values of hats is ranged 1-1001 so in this case 0 is a construct being applied and ultimately ignored because hat 0 is non-existent.
The problems are equivalent. The difference is the location of the additive identity.
Okay. Let's use 1-1001.
If you are adding the two numbers and subtracting 1001 if the sum is greater than 1001, then it is similar to modular arithmetic but it is not modular arithmetic. What you have basically done is move the additive identity from 0 to 1001.
If that is your method, then assume the hat no wizard is wearing is 999 and that the first three wizards are wearing hats 12, 10, and 1001 in that order.
If you are not adding the numbers as described above, then please explain your addition in detail.
I agree but it still seems as though the output would be exactly the same if the hats were numbered 0-1000 or 1-1001. The same uncertainty would apply.
They can't be exactly the same because the output would include a hat 1001, but no hat 0. The output from the same methods should be equivalent. If you are doing it like I think you are doing it, then your hat 1001 and my hat 0 are the same hat.
If the hats are numbered 1-1001, then you are not adding them together using modular arithmetic because in modular arithmetic, 1001 mod 1001 = 0. That is, you have to have a hat 0 and there cannot be a hat 1001.
That's why it makes sense to use the numbers 0 to 1000 instead of 1 to 1001 -- because you can then use modular arithmetic which is well understood.
Ahh. I get it now.
I only just learned about modular arithmetic when Somberlain posted the wikipedia so my understanding of it is still pretty basic. I was doing things the same way you suggested just with a different set. Thanks for teaching me something!
16 May 2013, 2:25 am
eric76 wrote:
aequitas1 wrote:
eric76 wrote:
aequitas1 wrote:
eric76 wrote:
Go back to this example:
Starting the index with hat 0 and using real modular arithmetic, if no wizard has hat 999 and the hat numbers of the first three wizards are 12, 10, and 0, what numbers do the first three wizards call?
Quote:
First, assume that the hats are numbered 0-1000 and that all additions are mod 1001.
Suppose that hat 999 is not being worn and that the first three wizards to guess have numbers 12, 10, and 0 in that order.
The first wizard sees that 999 and 12 are missing. So he adds them together to get 12+999 = 10 and says the number 10.
The second wizard doesn't see hats 10, 12, and 999 and concludes that his hat must be 10 but that it is already taken so he stays silent.
The third wizard doesn't see hats 0, 10, 12, and 999. The problem is that 0+10=10 and 12+999=10. What number is he going to call out?
Suppose that hat 999 is not being worn and that the first three wizards to guess have numbers 12, 10, and 0 in that order.
The first wizard sees that 999 and 12 are missing. So he adds them together to get 12+999 = 10 and says the number 10.
The second wizard doesn't see hats 10, 12, and 999 and concludes that his hat must be 10 but that it is already taken so he stays silent.
The third wizard doesn't see hats 0, 10, 12, and 999. The problem is that 0+10=10 and 12+999=10. What number is he going to call out?
Starting the index with hat 0 and using real modular arithmetic, if no wizard has hat 999 and the hat numbers of the first three wizards are 12, 10, and 0, what numbers do the first three wizards call?
While it is mathematically sound to construct a set beginning at 0, the only construction you can apply is cardinal assignments to the wizards but the riddle states that the values of hats is ranged 1-1001 so in this case 0 is a construct being applied and ultimately ignored because hat 0 is non-existent.
The problems are equivalent. The difference is the location of the additive identity.
Okay. Let's use 1-1001.
If you are adding the two numbers and subtracting 1001 if the sum is greater than 1001, then it is similar to modular arithmetic but it is not modular arithmetic. What you have basically done is move the additive identity from 0 to 1001.
If that is your method, then assume the hat no wizard is wearing is 999 and that the first three wizards are wearing hats 12, 10, and 1001 in that order.
If you are not adding the numbers as described above, then please explain your addition in detail.
I agree but it still seems as though the output would be exactly the same if the hats were numbered 0-1000 or 1-1001. The same uncertainty would apply.
They can't be exactly the same because the output would include a hat 1001, but no hat 0. The output from the same methods should be equivalent. If you are doing it like I think you are doing it, then your hat 1001 and my hat 0 are the same hat.
If the hats are numbered 1-1001, then you are not adding them together using modular arithmetic because in modular arithmetic, 1001 mod 1001 = 0. That is, you have to have a hat 0 and there cannot be a hat 1001.
That's why it makes sense to use the numbers 0 to 1000 instead of 1 to 1001 -- because you can then use modular arithmetic which is well understood.
They were the same hat. I really wish I learned the fundamentals of mathematics because I think I would have an easier time adjusting to the logical premises found more in the advanced fields of mathematics.
Is the expression 'mod 1001' always referring to a sequence that resets to 0 after 1000? So the expression 'mod 1001' refers to original arithmetic value of >1000? Why wouldn't it be expressed as 'mod 1000' instead of 'mod 1001'? Excuse my stupidity.
16 May 2013, 2:34 am
eric76 wrote:
aequitas1 wrote:
eric76 wrote:
aequitas1 wrote:
eric76 wrote:
Go back to this example:
Starting the index with hat 0 and using real modular arithmetic, if no wizard has hat 999 and the hat numbers of the first three wizards are 12, 10, and 0, what numbers do the first three wizards call?
Quote:
First, assume that the hats are numbered 0-1000 and that all additions are mod 1001.
Suppose that hat 999 is not being worn and that the first three wizards to guess have numbers 12, 10, and 0 in that order.
The first wizard sees that 999 and 12 are missing. So he adds them together to get 12+999 = 10 and says the number 10.
The second wizard doesn't see hats 10, 12, and 999 and concludes that his hat must be 10 but that it is already taken so he stays silent.
The third wizard doesn't see hats 0, 10, 12, and 999. The problem is that 0+10=10 and 12+999=10. What number is he going to call out?
Suppose that hat 999 is not being worn and that the first three wizards to guess have numbers 12, 10, and 0 in that order.
The first wizard sees that 999 and 12 are missing. So he adds them together to get 12+999 = 10 and says the number 10.
The second wizard doesn't see hats 10, 12, and 999 and concludes that his hat must be 10 but that it is already taken so he stays silent.
The third wizard doesn't see hats 0, 10, 12, and 999. The problem is that 0+10=10 and 12+999=10. What number is he going to call out?
Starting the index with hat 0 and using real modular arithmetic, if no wizard has hat 999 and the hat numbers of the first three wizards are 12, 10, and 0, what numbers do the first three wizards call?
While it is mathematically sound to construct a set beginning at 0, the only construction you can apply is cardinal assignments to the wizards but the riddle states that the values of hats is ranged 1-1001 so in this case 0 is a construct being applied and ultimately ignored because hat 0 is non-existent.
The problems are equivalent. The difference is the location of the additive identity.
Okay. Let's use 1-1001.
If you are adding the two numbers and subtracting 1001 if the sum is greater than 1001, then it is similar to modular arithmetic but it is not modular arithmetic. What you have basically done is move the additive identity from 0 to 1001.
If that is your method, then assume the hat no wizard is wearing is 999 and that the first three wizards are wearing hats 12, 10, and 1001 in that order.
If you are not adding the numbers as described above, then please explain your addition in detail.
I agree but it still seems as though the output would be exactly the same if the hats were numbered 0-1000 or 1-1001. The same uncertainty would apply.
They can't be exactly the same because the output would include a hat 1001, but no hat 0. The output from the same methods should be equivalent. If you are doing it like I think you are doing it, then your hat 1001 and my hat 0 are the same hat.
If the hats are numbered 1-1001, then you are not adding them together using modular arithmetic because in modular arithmetic, 1001 mod 1001 = 0. That is, you have to have a hat 0 and there cannot be a hat 1001.
That's why it makes sense to use the numbers 0 to 1000 instead of 1 to 1001 -- because you can then use modular arithmetic which is well understood.
I wasn't using proper modular arithmetic but the logic and strategy used was modular.
