[ninjinxo]

Since Peter is given the product of the two numbers, he should instantly know the pair if both numbers are prime, but since he doesn't it rules out pairs like (7x11) = 77 and (2x53) = 106.

Sandy knows the sum and has now been told that Peter doesn't know the pair. If the sum had been 6, the following pairs are possible: (1+5) (3+3) (4+2), Peter has just ruled out (1x5) and (3x3), so Sandy would be able to narrow it down to (1,5) if the sum had been 6.

So when she tells Peter she can't narrow it down, it tells him that the pair isn't (4,2) either (among many others).

And if Peter's number were 8: (1x8) or (2x4) he'd be able to solve it, but he doesn't so Sandy then knows that (1,8) isn't the solution either.

[LodeOfCode]

>he should instantly know the pair if both numbers are prime

That should actually be "if the product of their respective smallest prime factors is over 100". 7x11 can't be ruled out since 1x77 also produces 77, whereas 49x17 and Nx53 can be ruled out.

You can also rule out some of the larger squares, e.g. 25x25 and 64x64, so there's probably still a better phrasing for that

Edit: Can also rule out 1xPrime

[ninjinxo]

Oh whoops, good pickup; I'd like to say I tried to simplify it for the explanation and brevity, but I completely overloooked that 1xn = n.

[informal007]

>since he doesn't it rules out pairs like (7x11) = 77 and (2x53) = 106.

I think pair (7x11)=77 can't be rule out, because pair (1x77) is also equal 77.

still don't got it...

Can you explain the situation for 3 turns before Peter knows, Sincere thanks.

[indecisive_user]

It's easier if you think about a smaller range.

Let's think about picking two numbers between 1-9.

Peter is given the product 24. He knows there are two possible pairs of numbers between 1-9 which produce a product of 24, (3,8) and (4,6), so he says "I don't know the numbers"

Sandy is given the sum 10. There are many pairs of numbers that produce a sum of 10, [(1,9), (2,8)...]. But she also knows that Peter did not immediately know the answer. If the pair of numbers was (5,5), that would have produced a product of 25. If Peter was given a product of 25, he would have immediately known the answer, since there's only 1 pair of numbers that produces that product.

So Sandy knows the answer isn't (5,5). Similarly, she knows it's not (2,8) or (3,7). The answer could be (1,9) though, since the product of (1,9) is 9, and there's another pair that can produce that product (3,3). If Peter was given the product 9 he wouldn't have immediately known the answer. The answer could also be (4,6), since the product of those is 24, and that can also be achieved with the pair (3,8). So there's only 2 pairs of numbers that add up to 10, and which Peter would not have immediately known based on their product. Sandy knows the answer must be either (1,9) or (4,6). Sandy says "I don't know the numbers".

Peter knows the solution must be either (3,8) or (4,6), and he knows that Sandy did not immediately know the answer. If Sandy had been given the sum 11 though, she should have immediately known the answer. There is only 1 pair of numbers that produces 11, but which does not have a unique product. Yes, the pair (2,9) sums to 11, but the product is unique, and if Peter had been given the product 18 to begin with, he would have immediately known the answer. So because he didn't immediately know the answer, and because that was not enough information for Sandy to say that the pair is (3,8), then Peter knows that the summation of the numbers must not be 11. The only other choice then is (4,6), and so Peter says "I do know the numbers".

[informal007]

>So Sandy knows the answer isn't (5,5). Similarly, she knows it's not (2,8) or (3,7).

Why is not (2,8)? both pair (2, 8) and pair (4, 4) can produce a product of 16.

Thanks anyway!

[indecisive_user]

Yup, you're right, I missed that pair. But hopefully you get the general point on how it works :)

[informal007]

I get it from (this blog)[https://alexanderell.is/posts/numbers-game/], Thanks very much still!

[informal007]

sorry, I still can't get it...

> Yes, the pair (2,9) sums to 11, but the product is unique, and if Peter had been given the product 18 to begin with, he would have immediately known the answer.

pair (2,9) and pair (3,6) can produce a product of 18, so Peter can't immediately know the answer if he had been given the product 18 to begin with, so the problem can't beed solved.

I think this problem would Caught in an unresolved cycle after unique product and unique sum has been ruled out.

I don't know where i am wrong, Sincere thanks...

[herendin]

Probably easier if you imagine it with a much smaller range of numbers, like 1-9, or even 1-3