Alice and Bob
Facts
There are 2 mysterious integers A and B and two logicians Alice and Bob. Both integers are greater than 1 and less than 101. Alice is told the product of two and Bob knows the sum. Neither is told the values.
Conversation
Bob: I cannot determine a, b
Alice: I too cannot determine a, b
Bob: I already knew that.
Alice: In that case, I now know them
Bob: In that case, I too now know a, b.
How is this possible?
btw Happy thanksgiving
There are 2 mysterious integers A and B and two logicians Alice and Bob. Both integers are greater than 1 and less than 101. Alice is told the product of two and Bob knows the sum. Neither is told the values.
Conversation
Bob: I cannot determine a, b
Alice: I too cannot determine a, b
Bob: I already knew that.
Alice: In that case, I now know them
Bob: In that case, I too now know a, b.
How is this possible?
btw Happy thanksgiving
Labels: friday special, logic, puzzle
posted by Rajesh Lal at
9:41 AM
15 Comments:
Well i've never attempted one of these before lol. so i'm not all together sure really.. but running a few numbers in trial left me with only one conculsion to draw :-)
How is it possible? - They lied to each other!
I don't know about all of you, but i'm getting tired of the vagueness of these problems. i'm pretty sure someone out there is genius enough to come up with some equation where x = the product of the two and y = the sum and whatever...i don't know. but this is just beyond most people i know, because most people i know are not logicians, whatever they are. people who study logic, determine the obvious? the two numbers could be anything. 5 and 10. 2 and 7. 24 and 52. like really. what's the point? i think they were lying to each other.
Frustration is the point, however, the longer you do these and try them, the better you get at them. Logicians aren't born that way, they become that way...I happen to enjoy the problems and that most aren't just a quick computation problem... thanks for making us "think"....the dying artform...
Alice and Bob may be two mathameticians in the early stages of a degenerative disease such as Alzeihmer's. They are both suffering from conflabutatory amnesia. This occurs when the sufferer thinks they should know an answer but they don't... therefore they try to hide their ignorance by claiming to know something. For example Alice could ask Bob where his keys are and Bob would reply that he knew the location, even though he had forgotten and may not ever have known to begin with. Now. Solved.
Is it nine and 3? thought id just have a random guess.
I am Guessing too, but is it possible that ALice =A and Bob =B; therefore, the only way B can know A is if B=A??
One way this is possible:
Bob is told that the Sum is 11, so his possibilities are 2+9, 3+8, 4+7 and 5+6.
Since all of those pairs have at least 1 composite (not prime) number, he knows that Alice can't determine the product.
Alice is told that the Product is 18, so her possibilities are 3*6 and 2*9.
If the numbers were 3 and 6, then the Sum would be 9, and if they were 2 and 9, the Sum would be 11.
However, if Bob was told that the Sum was 9, he would know that the numbers could possibly be 2 and 7, and that the Product would be 14, and If Alice was told that the Product was 14, then she'd be able to determine that the numbers were 2 and 7 (the only prime factors of 14). But since Bob knows that Alice can’t determine the numbers, she now knows the Sum can’t be 9. Therefore, Alice knows the sum is 11 and that the numbers are 2 and 9.
Now that Bob knows that Alice knows the numbers, he can rule out 3+8, 4+7, and 5+6, all of which he knows that Alice could not deduce simply by knowing that he knows that she can’t deduce them, leaving 2 and 9 the only possibilities.
The solution is derived by knowing three things-
1)Bob knows the sum of A+B.
2)Alice knows the product of A*B.
3)Bob figured it out when he found that Alice didn't know.
If the integers were higher than 1 and lower than 11, then it would be known that integers include two of the numbers between 2 and 10.
If it were numbers 2 and 3 the product would be 6 and the sum would be 5. Breaking those two down show that the factors of 6 are 1,2,3,and 6.
The numbers are higher than 1 so factors 1 & 6 are illogical. But since we are dealing with ninety-nine integers instead of nine, it takes a bit more figuring. Especially since knowing the product would allow you to know, automatically, what the two are without knowning the sum.
If the integers were 20 and 52 the product would be 1040. The factors would be 2, 520; 4, 260; 5, 208; 8, 130; 10, 130; 13, 80; 16, 65; 20, 52; 40, 26.
Since the integers are between 1 and 101, the possible solutions would reduce to 13, 80; 16, 65; 20, 52; 40, 26.
Knowing the product alone would not allow the solution to be known, since those factors could be factors of other products as well. But knowing that the person who knows the sum cannot deduce the solution either allows you to know that the integers are factors of more than one product.
Narrowing it down to a sole set of factors which are factors of multiple products allow you to know the answer.
this is easy..they are both 2...one gets the product...one gets the sum...
they both get 4..and if it has to be over 1, than it must be 2 and 2.
100 bucks says that the numbers aare their ages.
The short answer is the numbers are 13 and 4
The solution is complicated though.
The third statement that Bob already knew Alice could not know the numbers means that the two numbers can not add up to a number which is the sum of two primes. If they did, the numbers might be the two primes themselves so Alice might have been able to deduce them right off the bat. E.g. if the sum is 12, the numbers might be 7,5 and alice would have been able to deduce them from the product of 35. For bob to say with certainty he already knew alice couldn't figure it out means the sum of the two numbers mustn't be the sum of two primes.
Creating a table of numbers (I only needed to go as high as 40) to figure out which numbers weren't the sum of two primes, you come up with
11, 17, 23, 27, 29, 35, 37. All the other numbers below 40 can be expressed as the sum of two primes.
Let's assume the sum is 11. It breaks down into four cases: 6+5, 7+4, 8+3, and 9+2
Case 1a) if the numbers were 6,5 the product would have been 30 and alice would think her possible solutions are 6,5 10,3 and 15,2. 6,5 was one candidate from Alice's point of view. She knows it couldn't be 10,3 because due to statement 3, their sum can't be the sum of two primes (10+3=13 which can be created by 11+2). She can't rule out 15,2 though as a possibility, therefore the numbers can't be 6,5. If they were, Alice could not have deduced them.
Case 1b) If the numbers were 7,4 alice would have the product 28 and have two possible solutions (7,4 and 14,2). Since 14,2 can be ruled out since it's sum is the sum of two primes, Alice would have been able to deduce that 7,4 were the numbers if she was given the product 28.
However, with both case 1c) 8,3 where she has the product 24 and case 1d) 9,2where she has the product 18 she would similarly **ALSO** have been able to deduce those numbers
This means there was more than one possible combination of numbers summing to 11 where alice could deduce what her numbers were (7,4 8,3 or 9,2). This means that B ob would never have been able to figure the numbers out at the end, based on the fact that Alice deduced them. Consequently the numbers can't sum to 11 and none of those candidates are solutions.
Case 2, the numbers sum to 17.
17 breaks down into 7 cases
9+8, 10+7, 11+6, 12+5, 13+4, 14+3, 15+2
Using the same process, Alice would be able to deduce the numbers only for one of the cases (13,4) but not for any of the other 6 of 7 cases. Consequently If bob is given the sum of 17 and alice is given the product of 52, they would conclude the numbers in exactly the sequence of statements identified.
I've broken into a mental sweat. Throw me a cerebral towel. Maybe a wet nap will do.
The short answer is the numbers are 13 and 4
The solution is complicated though.
The third statement that Bob already knew Alice could not know the numbers means that the two numbers can not add up to a number which is the sum of two primes. If they did, the numbers might be the two primes themselves so Alice might have been able to deduce them right off the bat. E.g. if the sum is 12, the numbers might be 7,5 and alice would have been able to deduce them from the product of 35. For bob to say with certainty he already knew alice couldn't figure it out means the sum of the two numbers mustn't be the sum of two primes.
Creating a table of numbers (I only needed to go as high as 40) to figure out which numbers weren't the sum of two primes, you come up with
11, 17, 23, 27, 29, 35, 37. All the other numbers below 40 can be expressed as the sum of two primes.
Let's assume the sum is 11. It breaks down into four cases: 6+5, 7+4, 8+3, and 9+2
Case 1a) if the numbers were 6,5 the product would have been 30 and alice would think her possible solutions are 6,5 10,3 and 15,2. 6,5 was one candidate from Alice's point of view. She knows it couldn't be 10,3 because due to statement 3, their sum can't be the sum of two primes (10+3=13 which can be created by 11+2). She can't rule out 15,2 though as a possibility, therefore the numbers can't be 6,5. If they were, Alice could not have deduced them.
Case 1b) If the numbers were 7,4 alice would have the product 28 and have two possible solutions (7,4 and 14,2). Since 14,2 can be ruled out since it's sum is the sum of two primes, Alice would have been able to deduce that 7,4 were the numbers if she was given the product 28.
However, with both case 1c) 8,3 where she has the product 24 and case 1d) 9,2where she has the product 18 she would similarly **ALSO** have been able to deduce those numbers
This means there was more than one possible combination of numbers summing to 11 where alice could deduce what her numbers were (7,4 8,3 or 9,2). This means that B ob would never have been able to figure the numbers out at the end, based on the fact that Alice deduced them. Consequently the numbers can't sum to 11 and none of those candidates are solutions.
Case 2, the numbers sum to 17.
17 breaks down into 7 cases
9+8, 10+7, 11+6, 12+5, 13+4, 14+3, 15+2
Using the same process, Alice would be able to deduce the numbers only for one of the cases (13,4) but not for any of the other 6 of 7 cases. Consequently If bob is given the sum of 17 and alice is given the product of 52, they would conclude the numbers in exactly the sequence of statements identified.
I've broken into a mental sweat. Throw me a cerebral towel. Maybe a wet nap will do.
why do you decide to answer these questions with large mathematical sequences when the answer is infront of your face. the answer is simply two. If Alice knows product and Bob knows sum and they are the same then the answer must be two because 2*2 and 2+2 equal the same number. FOUR.
First the list of possibilities where the sum =< 11
a=a b=b A=Alice(product) B=Bob(sum)
a b A B
2 3 6 5
2 4 8 6
2 5 10 7
3 4 12 7
2 6 12 8
3 5 15 8
2 7 14 9
3 6 18 9
4 5 20 9
3 7 21 10
4 6 24 10
2 9 18 11
3 8 18 11
4 7 24 11
5 6 30 11
Case a=2 an b=3
Bob gets the sum 5 and will know the answer !!
Case 2/4
Bob gets the sum 6 and will know the answer !!
2/5
Bob gets the sum 7
a/b can be 2/5 or 3/4
Alice gets the product 10 and will know the answer !!
2/6
Bob gets the sum 8
a/b can be 2/6 or 3/5
Alice gets the product 12
a/b can be 2/6 or 3/4
Bob reasons if a/b was 2/6 then Alice would get 12 and would not know
Bob reasons if a/b was 3/5 then Alice would get 15 and would know !!
2/7
Bob gets the sum 9
a/b can be 2/7, 4/5
Alice gets the product 14 and will know !!
2/8
Bob gets the sum 10
a/b can be 2/8, 3/7, 4/6
Alice gets the product 16 and will know the answer !!
2/9
Bob gets the sum 11
a/b can be 2/9, 3/5
Alice gets the product 18
a/b can be 2/9, 3/6
Bob reasons if a/b was 2/9 then Alice would get 18 and would not know
Bob reasons if a/b was 3/6 then Alice would get 18 and would not know
Bob reasons if a/b was 4/7 then Alice would get 28 and would not know
Bob reasons if a/b was 5/6 then Alice would get 20 and would not know
This could be an option for line 1, 2 and 3 !!<>!!
3/4
Bob gets the sum 7
a/b can be 2/5, 3/4
Alice gets the product 12
a/b can be 2/6, 3/4
Rob reasons if a/b was 2/5 then Alice would get 10 and would know. So he knows the answer !!
3/5
Bob gets the sum 8
a/b can be 2/7, 3/6, 4/5
Alice gets the product 15 and will know the answer !!
3/6
Bob gets the sum 9
a/b can be 2/7, 3/6, 4/5
Alice gets the product 18
a/b can be 2/9, 3/6
Bob reasons if a/b was 2/7 then Alice would get 14 and would know the answer !!
3/7
Bob gets the sum 10
a/b can be 2/8, 3/7, 4/6
Alice gets the product 21 and will know the answer !!
3/8
Bob gets the sum 11
a/b can be 2/9, 3/8, 4/7, 5/6
Alice gets the product 24
a/b can be 2/12, 3/8, 4/6
Bob reasons if a/b was 2/9 then Alice would get 18 and would not know
Bob reasons if a/b was 3/8 then Alice would get 24 and would not know
Bob reasons if a/b was 4/7 then Alice would get 28 and would not know
Bob reasons if a/b was 5/6 then Alice would get 20 and would not know
This could be an option for line 1, 2 and 3 !!<>!!
4/5
Bob gets the sum 9
a/b can be 2/7, 3/6, 4/5
Alice gets the product 20
a/b can be 2/10, 4/5
Bob reasons if a/b was 2/7 then Alice would get 14 and would know the answer !!
4/6
Bob gets the sum 10
a/b can be 2/8, 3/7, 4/6
Alice gets the product 24
a/b can be 2/12, 3/8, 4/6
Bob reasons if a/b was 2/8 then Alice would get 16 and would not know
Bob reasons if a/b was 3/7 then Alice would get 21 and would know !!
4/7
Bob gets the sum 11
a/b can be 2/9, 3/8, 4/7, 5/6
Alice gets the product 28
a/b can be 2/14, 4/7
Bob reasons if a/b was 2/9 then Alice would get 18 and would not know
Bob reasons if a/b was 3/8 then Alice would get 24 and would not know
Bob reasons if a/b was 4/7 then Alice would get 28 and would not know
Bob reasons if a/b was 5/6 then Alice would get 20 and would not know
This could be an option for line 1, 2 and 3 !!<>!!
5/6
Bob gets the sum 11
a/b can be 2/9, 3/8, 4/7, 5/6
Alice gets the product 30
a/b can be 2/15, 3/10, 5/6
Bob reasons if a/b was 2/9 then Alice would get 18 and would not know
Bob reasons if a/b was 3/8 then Alice would get 24 and would not know
Bob reasons if a/b was 4/7 then Alice would get 28 and would not know
Bob reasons if a/b was 5/6 then Alice would get 20 and would not know
This could be an option for line 1, 2 and 3 !!<>!!
Alice reasons if Bob knows that I don't know, he must have at least 11
If the product would be 18 she knows now that they are not 3 and 6 because the sum would be less then 11. So a and b are 2 and 9 or 9 and 2.
Frank R
Ok so I thought about it, and an integer, in math, is any number between -infinity & Infinity, but in reality it is just any complete Entity, any existence, so, i think the integers a&b are Alice and Bob, and their ages are the numbers between 1 and 101 so if they both say they cannot determine them, they both now know that their ages are the integers
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