Another Magic of Logic

Originally posted by BigDoggProblem
I just realized that I have not run the program since I set it to exclude sums with n1 or n2 greater than 100. Now it claims that (96, 99) and (97, 99) are solutions! I don't have time to analyze them by hand right now, but this is an interesting development nonetheless.

Could it be that the problem is unsound??

False alarm - these unusual cases (n1 and n2 both very close to 100) only revealed a minor programming bug. The cases of (98, 99) and (99, 99) almost lead to alternate solutions. Both can be summed only one way, because of the limit on n1 and n2. Therefore they lead to only one acceptable sum - but Mr. SUM would know n1 and n2 in advance because it's TOO narrowed down. I think the creator of the problem got a bit lucky here! (Especially if he assumed that all acceptable sums must be odd numbers....)

So, my earlier conclusions have not changed. I believe (4, 13) is the solution to the original problem (even though you can't assume all acceptable sums are odd, making the solution a bit less elegant).

I also still hold that the problem becomes unsound once you allows (437, 437) or higher.

Originally posted by BigDoggProblem
The problem with programs is that they're only too happy to show you an exception to a rule. It is false to say that there are no even acceptable sums. 174, 182, 184, 188, 190, 192, 196, and 198 are all acceptable. Why? Let's start with 174. The summing combinations can no longer start with something like 2+172, because 172 is too large for n1 or n2 ...[text shortened]... o bearing on the solution, but it shows how careful you must be in a problem of this complexity.

It is not correct to say that an even no. can be an acceptable sum.
It is a basic result of number theory, that any even number can be expressed as a sum of two primes. For example, to take your examples, for instance, we have clearly;
174 = 167 + 7.
182 = 179+3.
184=181+3
188=181+7,
190 = 179+11; 192= 179+13.

Originally posted by BigDoggProblem
Logical flaw? I sure hope you're not talking about limiting n1 and n2 to 99, because it was your suggestion to see what happens when n1 and n2 go up to 999. And now you claim your "contention stands correct". Hmm. That would be downright weasel-like behavior.

Again, please be specific. Which "logical flaw" are you referring to?

I have already conceded my error.

Originally posted by BigDoggProblem
False alarm - these unusual cases (n1 and n2 both very close to 100) only revealed a minor programming bug. The cases of (98, 99) and (99, 99) almost lead to alternate solutions. Both can be summed only one way, because of the limit ...[text shortened]... the problem becomes unsound once you allows (437, 437) or higher.

That is interesting. I stand corrected. Yes, even sums are also permitted if only 1<n1,n2<100 are to be considered. I withdraw my assertion in my last post. But that gives an interesting turn to the puzzle, for all those who might try to find some elegant way of finding its solution.
However ,I would like to make just one remark.
If SUM knows n1 and n2 even before PRODUCT's second statement, it mitigates against the very spirit of the puzzle, which is clear from the opening clause " That being so ......" of SUM's second statement.
This would imply that till the revelation of discovery (about his knowledge of n1 & n2) by PRODUCT, SUM had no idea of what n1 and n2 could be.

Originally posted by CoolPlayer
I have already conceded my error.

Sorry ... didn't mean to beat a dead horse. I was just curious what the rest of your post meant. You don't have to explain it if you don't want to.

Originally posted by ranjan sinha
If SUM knows n1 and n2 even before PRODUCT's second statement, it mitigates against the very spirit of the puzzle, which is clear from the opening clause " That being so ......" of SUM's second statement.

Right...The wording implies that Mr. SUM did not know n1 and n2 until his very last statement. That is why (98, 99) and (99, 99) are not solutions.

Originally posted by ranjan sinha

[B]By the way the logic applied by Plumber is not accurate.[/b]

Really? How so? (I don't recall stating any logic when I gave the answer - after you pointed out that n1 & n2 had to be greater than 1.)

Originally posted by The Plumber
Really? How so? (I don't recall stating any logic when I gave the answer - after you pointed out that n1 & n2 had to be greater than 1.)

I was only referring to your assertion about (4,13) being the "only" solution. This assertion was made without revealing the underlying fact of 1<n1,n2<100 being the limiting domain of consideration by both SUM and PRODUCT.

Of course I do admit , it is not so much a case
of inaccurate logic as one of incomplete logic . To that extent I do apologise. Sorry.

Originally posted by ranjan sinha
I was only referring to your assertion about (4,13) being the "only" solution. This assertion was made without revealing the underlying fact of 1<n1,n2<100 being the limiting domain of consideration by both SUM and PRODUCT.

Of course I do admit , it is not so much a case
of inaccurate logic as one of incomplete logic . To that extent I do apologise. Sorry.

Au contraire, mon ami....

Please check again, and you will see that my assertion of 4,13 was in response to your message stating that n1 and n2 were &quot;greater than unity.&quot; You had already stated that n1 and n2 were less than 100, hence 4, 13 was the only solution.

Once again, I should point out that I didn't state any &quot;logic&quot; in my post - I merely asserted that 4,13 was the only solution. Had I stated my &quot;logic&quot; then you could argue that it was incomplete or inaccurate. Without me stating my logic, I'm not sure how you can come to the conclustion that it is incomplete.

Originally posted by The Plumber
Au contraire, mon ami....

Please check again, and you will see that my assertion of 4,13 was in response to your message stating that n1 and n2 were "greater than unity." You had already stated that n1 and n2 were less than 100, h ...[text shortened]... ot sure how you can come to the conclustion that it is incomplete.

Je ne contraire pas, mon ami. Mais....pondrez si'il vous plait...


Are you trying to say that your assertion was without logic ?
Won't that make it illogical , which is worse than hidden logic?.

Originally posted by ranjan sinha
Are you trying to say that your assertion was without logic ?
Won't that make it illogical , which is worse than hidden logic?.

Of course he used logic. He was the first one to find the correct solution (4, 13) based on the correct stipulation. I think the odds of getting the right answer by chance or luck are pretty slim. He also said he used a spreadsheet, which again implies he knew what to put in it to find the answer.

I do not see any logical errors on ThePlumber's part. Without seeing his spreadsheet, how would you know whether he used good logic or not?

Originally posted by BigDoggProblem
Of course he used logic. He was the first one to find the correct solution (4, 13) based on the correct stipulation. I think the odds of getting the right answer by chance or luck are pretty slim. He also said he used a spreadsheet, which again implies he knew what to put in it to find the answer.

I do not see any logical errors on ThePlumber's part. Without seeing his spreadsheet, how would you know whether he used good logic or not?

Thanks. At least one person understood what I was trying to communicate....😉

Originally posted by BigDoggProblem
2) When SUM says "I knew you didn't know n1 or n2", I know that the sum can't be reached by adding prime + prime. I eliminate all sums with this possibility.

Why leave out sums with that possibility? The way i see it, every number has more then one possibilty of being a sum of two numbers.

let me give an example:

Why throw out 20 just because it's 17 + 3? The sum 20 is still valid at this point because it's also 16+4...or am i missing something (could very well be, since sleep is something i haven't had too much of in the last few days :p)

Originally posted by TheMaster37
Why leave out sums with that possibility? The way i see it, every number has more then one possibilty of being a sum of two numbers.
...
Why throw out 20 just because it's 17 + 3?

A sum of prime+prime leads to a product of prime*prime. Mr. Product would know n1 and n2 if the product is prime*prime. If there is even one possible prime+prime leading to a sum, Mr. Sum can't say with certainty that Mr. Product did not know n1 and n2.

Originally posted by BigDoggProblem
Of course he used logic. He was the first one to find the correct solution (4, 13) based on the correct stipulation. I think the odds of getting the right answer by chance or luck are pretty slim. He also said he used a spreadsheet, ...[text shortened]... spreadsheet, how would you know whether he used good logic or not?

True . Plumber was the first to come up with the correct answer, albeit , without clearly spelling out the logic behind his assertion or the finding. I never assumed his assertion was without logic. Only he said that he didn'd give any logic. In any case one could make valid assumption about the implicit ( hidden) logic employed by him. It was you who also found out the multiple solutions e.g. (4,61), (16,73) within the 1&lt;n1,n2&lt;100 range, contradicting his (Plumber's) assertion.

It was at that point that the thus far ignored fact of only the limiting domain of n1, n2 to be considered by both SUM and PRODUCT (while examining the possible factorisatin for the known product by PRODUCT and for the possible break up of the known sum by SUM), was pointed out by me for the first time.
Plumber did not react to your findings , contradicting his assertion. Though he had come up with the correct answer , through his undisclosed or implicit logic , he could not give the clarification why the alternative answers pointed out by you were not correct solutions of the problem.

It was not unreasonable to assume that Plumber had possibly ignored the above factor, (as pointed out above), in his analysis, otherwise, he ought to have reacted as soon as you disclosed the other solutions , found in your computer search. .