8
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My friends Peter and Sam are excellent mathematicians and always think strictly logical. Yesterday I told them: "I have secretly chosen two integers $x$ and $y$ with $1\le x\le y\le 9$. I have told their sum $s=x+y$ to Sam and their product $p=xy$ to Peter." Then the following conversation developed.

  1. Peter said: I don't know the numbers.
  2. Sam said: I don't know the numbers.
  3. Peter said: I don't know the numbers.
  4. Sam said: I don't know the numbers.
  5. Peter said: I don't know the numbers.
  6. Sam said: I don't know the numbers.
  7. Peter said: I don't know the numbers.
  8. Sam said: I don't know the numbers.
  9. Peter said: Aha. Then I do know the numbers now.
  10. Sam said: Aha. Then I also know the numbers now.

What are the values of $x$ and $y$?

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6
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To solve this I listed all possible solutions and then removed each entry that couldn't be it after the question. The thing is, a person knows what value x and y if there is only one entry in that list with that given s or p. Saying that you don't know the solution implies that I can remove all items from the list that only have a single entry with that s or p. Here is the list:

xy  s  p
11  2  1
12  3  2
13  4  3
14  5  4
15  6  5
16  7  6
17  8  7
18  9  8
19 10  9
22  4  4
23  5  6
24  6  8
25  7 10
26  8 12
27  9 14
28 10 16
29 11 18
33  6  9
34  7 12
35  8 15
36  9 18
37 10 21
38 11 24
39 12 27
44  8 16
45  9 20
46 10 24
47 11 28
48 12 32
49 13 36
55 10 25
56 11 30
57 12 35
58 13 40
59 14 45
66 12 36
67 13 42
68 14 48
69 15 54
77 14 49
78 15 56
79 16 63
88 16 64
89 17 72
99 18 81

After question 1 I can remove 11, 12, 13, 15, 17, 25, 27, 35, 37, 39, 45, 47, 48, 55, 56, 57, 58, 59, 67, 68, 69, 77, 78, 79, 88, 89 and 99.

That leaves us with:

xy  s  p
14  5  4
16  7  6
18  9  8
19 10  9
22  4  4
23  5  6
24  6  8
26  8 12
28 10 16
29 11 18
33  6  9
34  7 12
36  9 18
38 11 24
44  8 16
46 10 24
49 13 36
66 12 36

After question 2 we remove 22, 49, 66. we now have:

xy  s  p
14  5  4
16  7  6
18  9  8
19 10  9
23  5  6
24  6  8
26  8 12
28 10 16
29 11 18
33  6  9
34  7 12
36  9 18
38 11 24
44  8 16
46 10 24

After question 3 we eliminate 14, we now have:

xy  s  p
16  7  6
18  9  8
19 10  9
23  5  6
24  6  8
26  8 12
28 10 16
29 11 18
33  6  9
34  7 12
36  9 18
38 11 24
44  8 16
46 10 24

After question 4 we eliminate 23, we now have:

xy  s  p
16  7  6
18  9  8
19 10  9
24  6  8
26  8 12
28 10 16
29 11 18
33  6  9
34  7 12
36  9 18
38 11 24
44  8 16
46 10 24

After question 5 we eliminate 16, we now have:

xy  s  p
18  9  8
19 10  9
24  6  8
26  8 12
28 10 16
29 11 18
33  6  9
34  7 12
36  9 18
38 11 24
44  8 16
46 10 24

After question 6 we eliminate 34, we now have:

xy  s  p
18  9  8
19 10  9
24  6  8
26  8 12
28 10 16
29 11 18
33  6  9
36  9 18
38 11 24
44  8 16
46 10 24

After question 7 we eliminate 26, we now have:

xy  s  p
18  9  8
19 10  9
24  6  8
28 10 16
29 11 18
33  6  9
36  9 18
38 11 24
44  8 16
46 10 24

After question 8 we eliminate 44, we now have:

xy  s  p
18  9  8
19 10  9
24  6  8
28 10 16
29 11 18
33  6  9
36  9 18
38 11 24
46 10 24

Conclusion:

If Peter didn't know it now, he would eliminate 28 now which means that x = 2 and y = 8 is the answer. Sam knows this too because he thinks the same like Peter (and me).

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