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If Sam had the product and Pru the sum I can see an answer (13 & 4)

But it is the other way round. I can find a way to get 9 & 2, but the argument is not convincing.

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It’s good that we’ve got something to work on. Your suggestion that roles may have been reversed had me a bit worried that I might have written the question the wrong way round (always a possibility). Let’s explore it:

Hopefully this isn’t too much of a hint for others but the first statement is essentially saying: “based on what I have, I know you have more than one possible pairing”. But that’s true of nearly all the sums. I think in fact all of them apart from those at the boundaries, the only immediately solvable sums are: 2+2, 2+3, 19+20 and 20+20 and in such cases the product is immediately solvable too. So how does the statement allow the sum to be determined? Of course I may be missing something in which case there could be a variant of the puzzle on the lines you have indicated. As Don said, there are variations of the puzzle out there.

Your possible solution of 9 and 2 with the question as worded is worth exploring. I’ll try not to give too much away:

The sum of 11 is consistent with the first text. It is also consistent with the second text in that it would allow Pru to solve the puzzle. But I think it leaves three possible solutions being 2x9, 3x8 and 4x7 (I won’t explain just yet but 5x6 can be ruled out), so Sam wouldn’t be able to solve it in the final step.

So it looks like you are pretty much on the right lines. Given that you have got so far it may be worth just double checking the application of your logic.

I don’t think I have the time to devote, at least not unless I suddenly have a brainwave,

Sure, I understand.

At least you must have got over the first big hurdle. That is, realising it is deliberately set out to appear impossible, yet is ultimately solvable. Even though bright students such as Sam and Pru have more information than we do, it is impossible for either of them to solve it without the seemingly innocuous text messages.

If anyone wants to set out their thoughts, even for a partial solution, that would be good. I am sure Don will chip in or set out his solution at some point. I will wait until then. Though if anyone is trying to solve this and wants some hints to progress, please let me know.

I am thinking that the two numbers must be prime numbers.

Prime numbers are a key part of the solution so it is worth developing thoughts along those lines. However if they were both prime then Pru would have known the answer before Sam texted her. The product of two primes has only one answer, e.g. 21 can only arise from 3x7 etc. That feature of prime numbers is something that can be used.

Dozey,
The numbers can’t be prime, for the reason Ravvie said.

Nor can they produce a unique product eg 2x4=8 is the only way in which the product 8 can be generated and Pru would have been able to solve the problem immediately if the product was unique in this respect.

Then the converse can be used - I.e they cannot both be primes.

That’s correct Dozey.

And the combination can’t be both 2 and 4, but it doesn’t exclude both 2 and 4.

Would 3 and 4 work? On second thoughts Sam would get the answer immediately as the sum of 7 would only give products 12 and 10, and if it was 10 Pru would get the answer immediately.

Well, if Sam had the sum 7, he would know the product could only be 2x5=10: or 3x4=12. But he wouldn’t know which.

Equally, Pru, who would have the product 12, would know that Sam could have a sum 6+2=8 or 3+4=7. But again, she wouldn’t know which.

It would be worth trying to generalise your example. You have ruled it out (correctly) because 2+5 would mean Pru solved it already so if the sum is 7 then Sam can’t be sure that Pru can’t have solved it.

After the texts in bold, I suppose I should have added something like "until they started talking"

BTW, just to avoid “red herrings”, 3 and 4 don’t work as a solution.

Sam would know it wasn’t 2x5 because then Pru would have the answer.

Yes, that’s a really important step in the logic.

You are right about the 2x5=10 Dozey.

Best to go back to the link I have made with my earlier post.

Sam can eliminate all the prime numbers from the list of Products and also combinations that produce a unique Product, such as 2x4=8 and, as you correctly point out 2x5=10 is another example of one of these unique Products. For sure, if Pru had on of these numbers as her Product, she would have been up and finished in a flash !

When I first encountered this teaser a few years back, I drew up a chart with the possible Sums along the top and the possible Products down the left hand side. It was tedious, But helped me to cross-check all the possible combinations of Sums and Products.

You might like to give it a try. I think you could limit the combined Sum to 20 instead of 40, just to reduce the tediousness. I think this will give a unique solution, and quite probably the same solution as per Ravvie’s teaser with 40 as the upper boundary.

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Well, I worked out that Sam’s possible sums are 11, 17, 19, 23 or 37. As you say it is less than 20 that gives me 11, 17 or 19 to work with. Now need to look at products for these sums. And we know from above that 11 doesn’t work, which leaves sums of 17 or 19.

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I think the sum might be 17 and the product 72. The numbers thus being 8 and 9.

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Dozey,

I think you are getting really close. But please check your possible sums. For example, 19=2+17, so I think should be excluded since Pru would solve 2x17 straightaway.

There are also some possible sums between 20 and 37 that I think should be included. I know Don suggests limiting the sum to 20, but I don’t follow why. I chose 40 as the limit because I needed sums up to 37 to be potentially included. Of course, Don may be using a different method to arrive at the answer, but there should only be a unique answer. I don’t want to give too much away but I rule out 8 and 9.

Maybe @Don could comment?