Hey, here's one that I recently found very puzzlingly enjoyable;
Two whole numbers, m and n, have been chosen. Both are unequal to 1 and the sum of them is less than 100. The product, m × n, is given to mathematician X. The sum, m + n, is given to mathematician Y. Then both mathematicians have the following conversation:
X: "I have no idea what your sum is, Y."
Y: "That's no news to me, X. I already knew you didn't know that."
X: "Ahah! Now I know what your sum must be, Y!"
Y: "And now I also know what your product is, X!"
The Question: What are the numbers m and n?
Two whole numbers, m and n, have been chosen. Both are unequal to 1 and the sum of them is less than 100. The product, m × n, is given to mathematician X. The sum, m + n, is given to mathematician Y. Then both mathematicians have the following conversation:
X: "I have no idea what your sum is, Y."
Y: "That's no news to me, X. I already knew you didn't know that."
X: "Ahah! Now I know what your sum must be, Y!"
Y: "And now I also know what your product is, X!"
The Question: What are the numbers m and n?
-
Re: sum and product puzzle
Wed, March 1, 2006 - 5:33 AMX: "I have no idea what your sum is, Y."
The product is not the product of two primes.
Y: "That's no news to me, X. I already knew you didn't know that."
The sum is not the sum of two primes.
X: "Ahah! Now I know what your sum must be, Y!"
Only one of the sums that gives this product is both less than 100 and not the sum of two primes
Y: "And now I also know what your product is, X!"
Only one product has this property.
Karl -
-
Re: sum and product puzzle
Thu, March 2, 2006 - 4:49 AMI thought about this again and realised that my last translation was not correct. I repeat the translations with correction.
I've also put the possesives into the translation to remind you that X knows the product of the two numbers and Y knows the sum.
X: "I have no idea what your sum is, Y."
My product is not the product of two primes.
Y: "That's no news to me, X. I already knew you didn't know that."
My sum is not the sum of two primes.
X: "Ahah! Now I know what your sum must be, Y!"
Only one of the sums that gives my product is both less than 100 and not the sum of two primes
Y: "And now I also know what your product is, X!"
Only one product for my sum has this property.
Karl
-
-
Re: sum and product puzzle
Wed, March 8, 2006 - 6:10 AMI think 4 and 13 could be those numbers:
X: "I have no idea what your sum is, Y."
Your sum could be 17 or 28
Y: "That's no news to me, X. I already knew you didn't know that."
17 is not the sum of two primes. Hence every possible product fails to tell you the sum.
X: "Ahah! Now I know what your sum must be, Y!"
28 is the sum of two primes (5+23 or 11+17) so has products that tell you the sum.
Y: "And now I also know what your product is, X!"
17 has the following possible products (30, 42, 52, 60, 66, 70, 72) of which only 52 is unique to this sum of 17, so it must be 52.
One outstanding question:
Is there a simple proof that this is the only solution
or is it necessary to check many numbers? -
-
Re: sum and product puzzle
Wed, March 8, 2006 - 1:43 PMGood question. I took the trusting route... I figured out the rules like you did, and then when I found two numbers that satisfied them I just accepted the fact that they were the only answer and stopped looking any further!
-
Re: sum and product puzzle
Mon, March 10, 2008 - 1:27 PMKarl, how long did it take you to solve this problem? And do you think it would tremendously more difficult without a calculator? (I'm asking for the same reason I asked about the planets problem — because I may use it for a contest I'm running, and I can't remember how long it took me. There's a time limit to the contest. And the puzzles need to be solvable without calculators.) thanks!
-
