Logic
Amanda, Billy, Caleb, David, and Ellie are told that they are each given a distinct integer from 1 to 5 inclusive. They each know their own integer, but are not told the integer of anyone else. They make the following statements:
Amanda
: "My number has an odd number of positive factors."
Billy
: "Really? My number is either odd or prime, but not both."
Caleb
: "I now know Amanda's number."
Given that David's number is less than Amanda's number, what number does Ellie have?
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8 solutions
Moderator note:
Would the solution to the problem be different if Caleb's statement came before Billy's?
The final solution would be the same, but Caleb would have 1 and Billy would have 2... in a way I find this version of the question easier:)
How did we deduce that Caleb deduces Amanda's number only after Billy's statement and not just Amanda's statement?
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Ahh, I see what you mean. It's my interpretation on the wording of the question. He says "I now know", which implies that he needed the statements from both people to work it out.
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Caleb would have to have 1 in order to deduce Amanda's number immediately after her statement. However, it doesn't change the answer being sought.
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@Rob Friars – Good point! I didn't notice that. David's and Ellie's numbers remain the same. However, we cannot determine between Amanda and Caleb who has 1 and who has 4.
I always thought 1 was prime. Learn something new every day.
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It is (basically)--and, it should be noted the proof functions regardless of whether unit is treated as prime or not.
The main reason 1 is not "treated" as prime is due to a shorthand abbreviation (read: congnitive laziness) in arithmetic operations over real numbers--i.e. the specific attributes of primality get "problematic" when considering the unit value for multiplication (over the Complex numbers) [somewhat corollary to the problem of calling 0 "odd"].
While I am not in "sync" with the predominant thought on the subject (at least outside of Analytic circles), I would argue 1 (the value) IS prime--it is divisible only by the metric unit and itself. However, when not evaluating 1 (i.e. as a value), one only need remember 1 (like zero for addition) is NOT a factor of relevance besides its use as an "inert" reference point (i.e. as unit).
[The easiest corollary would be for addition--the fact that 0 is even derives from its ontology, but the fact that "even + odd = odd" when the even is 0 does not; instead it derives from the fact that adding zero (as the origin) does nothing to the value. Just because it "behaves" like something, doesn't prove it belongs...and vice versa, of course!]
No, but it is possible the problem would no longer be solvable as written. If Caleb knew Amanda's number after her statement it would be possible for Caleb to have 4 and Amanda to have 1, in which case it could no longer be given that David's number is less than Amanda's. Hence they rely on the logic puzzle trick of statement order and the use of the word 'now' to make it work.
No. The only thing that would have changed were Caleb and Billy's switching numbers:
Amanda could still only have either the numbers 1 or 4
So if Calebs statement came next that would mean that he had either the number 1(if Amanda has 4) or the number 4 (if Amanda has 1)
This leaves the numbers 2, 3, and 5 left,
Because Billy's number "is either odd or prime , but is not both", that means that Billy must have the number 2 (as the numbers 3 and 5 are both odd AND prime numbers)
In order for David to have a number lower than Amanda's, she cannot have the number 1. Thus, she must still have the number 4,
This would also mean that Caleb much have the number 1
And that still leaves Ellis with the number 5
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So the new order would be:
Caleb has the number 1
Billy has the number 2
David has the number 3
Amanda has the number 4
And Ellis has the number 5
We can arrive at the answer without first deducing Caleb's number, or indeed Amanda and Billy's numbers.
Amanda = 1 or 4, Billy = 1 or 2. So far the solutions could be (1,2) (4,1) or (4,2). If Caleb = 3 or 5 he would not be able to know which of those solutions apply. Therefore Caleb = 1, 2 or 4. Therefore David and Ellie = 3 and 5 between them. But David's number is less than... (we need read no further). David is not 5. So Ellie is 5.
Relevant wiki: K-level thinking
First, we know that the only numbers that have an odd number of factors are 1, which has only itself, and 4, which has 1, 2, and 4. This means Amanda has either a 1 or a 4 as her number.
Second, lets look at Caleb's number (You'll see why shortly), The only way he would know Amanda's number based on what she said is if he also had a number with an odd number of factors as well, as there are only two of them. If Amanda had 4, he would have 1, and vice versa.
Next, let's have a look at Billy's statement. If his number is either odd or prime, it can't be both, which crosses out 3 and 5. It can't be 1 or 4, which leaves him with the number 2. We now know that David and Ellie have the numbers 3 and 5 in some order.
Now, we know that David's number is less than Amanda's number. If Amanda's number were one, this would not be possible because one is the smallest number among them. Therefore, Amanda's number must be 4 and Caleb's number must be 1. David's number can't be 5 because it would be greater than all the numbers, so he has the number 3. This leaves Ellie with the number 5 .
Summarizing, Caleb has 1, Billy has 2, David has 3, Amanda has 4, and Ellie has 5.
Moderator note:
In logic problems, the choice of words could have serious implications on the sequence of logical deductions. Especially in level k thinking, we have to be careful to verify why each person knows at each given instant in time. Switching up the order can lead to a different sequence (and hence result).
@Christopher Ho Thinking about this further, I believe that your solution misses our a possible scenario.
We are given that:
1. Amanda has 1 or 4.
2. Billy has 1 or 2.
3. Caleb
now
knows Amanda's number.
Especially with the word now , that suggests that he needs Billy's statement to make the conclusion. This suggests to me that Caleb doesn't have 1 or 4. Instead, Caleb has a 2, and now he knows that Billy has a 1, and hence Amanda has a 4.
Luckily, it doesn't change the final answer :)
Edit: I see that Dan raises this possibility in his solution.
Why can't Billy have chosen 1? The number 1 is odd, but not prime.
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While it's true that based on Billy's statement in isolation his number could be 1 or 2 , since we know Amanda and Caleb have the numbers 1 and 4 in some order we can conclude that Billy's number can't be 1 , and thus must be 2 .
@Christopher Ho - Could you edit the statement "David's number is less than Amanda's number " to "David's number is less than Ellie's number".
Else, we could guess Amanda's number right away as there is no number less than 1 , making Amanda's number 4 .
Billy Could have chosen 1
I see now that your "either ... or" statement is exclusive. I took it as inclusive, so when Billy says that his number is either odd or prime, I interpreted this to mean that his number is either odd or prime or both odd and prime, which would in turn mean that his number can be any of 1 , 2 , 3 or 5 . With this interpretation there would not be enough information to determine Ellie's number, (or Billy's or David's either). The default for the "either ... or" construct often depends on context, and while your wording is probably just fine, for sake of absolute clarity it may be worth editing Billy's statement to "Really? My number is either odd or prime, but not both."
P.S.. Congrats on posting your first question.
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Thanks, I've edited the problem for clarity.
I was thinking Billy's number is either 1 or 2 because 1 isn't prime and two isn't odd yet it's prime.
I also saw the "either or" statement the same easy and thought there wasn't enough info.
Christophe,r I'm very glad you realized that the problem as stated places a great burden on the word "now". Just because I know something "now" doesn't mean I didn't also know it long ago. As stated there is no reason Caleb might not have a "1" and knew that Amanda had a 4 immediately after her statement. Perhaps he tried to say so, but Billy was just faster than him at speaking up.
"My number has an odd number of positive factors." what does it mean !!!!
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Consider 25. It's factors are 1, 5 and 25. There are 3 factors of 25, so 25 has an odd number of positive factors.
Consider 6. It has an even number of positive factors.
First Amanda says she has an odd number of positive factors. 1 has only itself as a positive factor, so 1 is a possibility, and there is also 4 since it has the positive factors of 1,2 and 4
So Amanda has: 1 or 4
Billy says my number is odd or prime but not both, only 1 is odd and not a prime, and 2 is a prime but not odd
So Billy has: 1 or 2
Then after this piece of information Caleb finds out his number, which must be 2 if he know knows he can eliminate 1 and know Amanda's number
So we know now: Amanda has 4, Billy has 1, Caleb has 2
Then it says David's number is less than Amanda's and the only possible number left is 3, so:
Amanda has 4, Billy has 1, Caleb has 2, David has 3
The only number remaining is 5 which goes to Ellie, which makes your answer 5 (:
Ah, this is a very systematic approach! It's very clear from your explanation why we can easily eliminate the other options one at a time.
Amanda's number is a square, so her number is either 1 or 4.
We're given that David's number is less than Amanda's number: therefore Amanda's number is 4.
We're given no further information about David or Ellie's number. However, we also know that this is enough information to solve the problem. Therefore Ellie's number must be greater than Amanda's number (otherwise there would be no way to know which was David's and which was Ellie's).
Ellie's number must be larger than 4: therefore it is 5.
Not a solution, but an offshoot of Amanda's situation. Probably my favorite math puzzle, cuz the reason for the conclusion is so cool.
There is an infinitely long corridor of numbered lights (1, 2, 3, 4, 5, ...) with pullchains. Initially all lights are off.
Person #1 goes down the corridor and pulls all the chains.
" #2 pulls #2, #4, #6, #8, .... ad infinitum
" #3 " #3, #6, #9, #12, ... "
" #4 " #4, #8, #12, #16, ... "
And this routine continues with an infinite number of persons (or anyway, as many as you want)
When all is done, which lights are on?
This is indeed a nice puzzle! How about you share this as a problem so others will be able to solve it and discuss its solutions?
I think this question that you pose is a math lore. Let me guess, the answers are a certain perfect powers, right?
Billy has the number 1
Caleb has the number 2
David has the number 3
Amanda has the number 4
This means that Ellis has the number 5
Let's look at each statement carefully:
Amanda: "My number has an odd number of positive factors."
--- The only number that have an ODD number of positive factors are perfect squared numbers. In this case, this would be number 1 or 4.
Billy: "Really? My number is either odd or prime, but not both."
--- This is either 1 or 2 , as 1 is an odd number but not prime a and 2 is prime number but not odd . ( Also, 3 and 5 are both prime AND odd, and 4 is neither odd nor is it prime. Therefore, Billy could never have the numbers 3, 4 or 5.)
Caleb: "I now know Amanda's number."
--- This means that Caleb has the number 2. In order for Caleb to know Amanda's number, he would need to know Billy's number first.
--- If Caleb had the numbers 1 or 4, he would have figured out Amanda's number in the first statement.
--- If Caleb had the numbers 3 or 5, he would not have been able to figure out Amanda's number after the second statement.
--- But if Caleb has the number 2, he would then know that Billy has the number 1, and therefore Amanda must have the number 4.
This now leaves the numbers 3 and 5.
And given the following statement:
"David's number is lower than Amanda's"
--- David's number must be the number 3, (Amanda has the number 4 and 3 is the only number left that's less than 4)
This leaves Ellis with the number 5
Why is it possible to Billy to have the nunber 1? 1 isn't prime AND odd?
If one were prime then the fundamental theorem of arithmetic wouldn't make sense, as it is.
An alternative explanation is:
A prime number has only 2 distinct divisors, 1 and itself.
So, by definition, 1 is not a prime.
Wow! I can not believe that I got it correct by randomly pressing an answer
And I got it wrong after careful and correct analysis.
Firsts note that the factors of 4 are 2 and 2. If we want to include 1, then the factors of 4 might be 1 * 1 * 2 * 2. We were not specified unique factors (in which case the factors of 4 are 1 and 2 by this logic) or non redundant factors.
While it is true that the divisors of 4 are 1, 2, and 4 I don't think that is what they said.
Amanda has 2,3 or 5 - the only primes and 4 = 2 * 2. Units don't factor
Billy has 2. 1 is not odd, but 1 is not prime either. So Amanda has 3 or 5.
Caleb knows that Amanda has 3 or 5 because he has the other one.
David's number is less than Amanda's. So there are two possible solutions.
A3 B2 C5 D1 and E 4
or
A5 B2 C3 D 1 or 4 and then E 4 or 1
Given the problem has a solution it must Ellie must have 4.
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You might want to review your definition of 'correct'.
The positive factors of 4 are definitely 1, 2 and 4 (if the problem says 'the factors,' your expectation is a reasonable list, not one random multiplication). A proper divisor is synonymous to a factor (factor is simpler and more well understood). → Amanda either has 1 (=1) or 4 (=1×2×4). (The other numbers are prime, yes (1×2, 1×3, 1×5): they have 2 factors by definition, an even number, so not what Amanda has. Only square numbers have an odd number of factors – because factors come in pairs unless they are multiplied by themselves.)
1 is definitely odd. There are no exceptions to the rule that even numbers are multiples of 2 and odd numbers are not. → Billy either has 1 (odd but not prime) or 2 (prime but not odd).
Then, armed with correct maths, you can go into the actual solution, which I'm sure you've already read, and which I can't explain as well as has already been done.
From the information given:
Amanda : Number 1 or 4
Billy : Number 1 or 2
Caleb does not know after Amanda 's statement whether she has number 1 or 4 . However, after Billy states that he has number 1 or 2 , Caleb knows the number Amanda has. This means that Billy has a number that excludes one from Amanda 's (which is 1 , the only number they have in common). And the only way that Caleb could have known that Billy has number 1 is if he himself has number 2 .
David : Since Amanda has number 4 , Billy has number 1 and Caleb has number 2 , David must have number 3 , as his number is less than Amanda 's.
Ellie has the only number remaining, which is 5 .