Determine the number $r$ from the statements

Question

Alice and Bob are perfectly logical and super intellegent. Their professor decides to play a game with them, and he tells them: "I have chosen two numbers (integers) $x$ and $y$ with $2\le y \le x\le 100$. I will tell Alice the value of their difference $d=x-y$ and Bob the value of their ratio $r=x/y$. I stress that the ratio $r$ is an integer, too."

After the professor has told the difference to Alice and the ratio to Bob, the following exchange occurs:

1. Alice: I don't know the numbers.
2. Alice: You might though. Do you know them?
3. Bob: No, I don't know them.
4. Alice: Too bad, if you did I would know them too.
5. Alice: I still don't know them though.
6. Bob: Me neither.
7. Alice: Oh really? Then I do.
8. Bob : Dang it; I still don't.

What is the value of $r$?
What are the possible values for $x$ and $y$?

Answer 1

This is how I reasoned:
There are 382 sets with 2 ≤y≤x≤100 where x/y is an integer.

Statement 1 says Alice doesn't know the numbers.
This means that we can eliminate all sets that have a difference d that only appears once.
357 remain.

Statements 2 & 3 say that Bob doesn't know them,
so we can eliminate sets that have a ratio r that only appears once.

Statement 4 however says, if it had been one of those sets, then Alice would know the numbers, because in all those sets, y=2.
So d must be one of the d's that appeared in one of the sets just eliminated.
Only 49 other sets have one of those d's.

5: Alice still doesn't know the numbers, so we can again eliminate sets with a unique d.
46 remain.

6: Bob still doesn't know, so we can eliminate sets with a unique r.
34 remain.

7: Now, suddenly Alice knows the numbers, so d must be unique among the remaining sets.
Only 4 sets remain:
{x=85, y=17, r=5, d=68}
{x=95, y=19, r=5, d=76}
{x=91, y=13, r=7, d=78}
{x=100, y=4, r=25, d=96}

8: Bob still doesn't know x and y, so r is still not unique.
Only 2 sets remain:
{x=85, y=17, r=5, d=68}
{x=95, y=19, r=5, d=76}
So r is 5 and x,y is either 85,17 or 95,19. Alice knows, but we don't.

Answer 2

The values are:

r = 5, d = 76 or 68, so x,y = 95,19 or 85,17

Reasoning:

Step 1 tells us there must be multiple solutions for d. This rules out large primes such as d=89, and also numbers like d=91 (which must have x,y=98,7 and r=14).
Step 3 tells us that r <= 32, because otherwise the only possible value for y would be 2 and Bob would know the solution.
Step 4 tells us that r could have been >= 33, otherwise Alice would not have had a solution if it was. So d must be >= 66.
Step 5 further limits the solutions. Suppose by way of example, that d = 92. Then x,y,r must be 94,2,47 or 96,4,24. If d were 92, Alice would now know the answer because r < 33, but she does not. We can eliminate all solutions with r >= 33.
Step 6 eliminates more solutions: for example d=84 could mean x,y,r = 84,4,22, and this is the only solution with r=22 and d >= 66. Since Bob still doesn't know the answer, it can't be 84, and r can't be 22.
Step 7 means that the solution must be a d value where only one of its possible solutions shares an r-value with another d-value. The only way to find these is to write them all down, and they are d,r=96,25; 92,24; 76,5; 68,5.
Step 8 means that the solutions must share r values, or Bob would know which one to choose, so r=5 and d=76 or 68.

Answer 3

first I list all possible solutions that satisfy 2 <= y < x <= 100 where y is a divisor of x

x   y   d   r
100 50  50  2
98  49  49  2
96  48  48  2
94  47  47  2
92  46  46  2
90  45  45  2
88  44  44  2
86  43  43  2
84  42  42  2
82  41  41  2
80  40  40  2
78  39  39  2
76  38  38  2
74  37  37  2
72  36  36  2
70  35  35  2
68  34  34  2
99  33  66  3
66  33  33  2
96  32  64  3
64  32  32  2
93  31  62  3
62  31  31  2
90  30  60  3
60  30  30  2
87  29  58  3
58  29  29  2
84  28  56  3
56  28  28  2
81  27  54  3
54  27  27  2
78  26  52  3
52  26  26  2
100 25  75  4
75  25  50  3
50  25  25  2
96  24  72  4
72  24  48  3
48  24  24  2
92  23  69  4
69  23  46  3
46  23  23  2
88  22  66  4
66  22  44  3
44  22  22  2
84  21  63  4
63  21  42  3
42  21  21  2
100 20  80  5
80  20  60  4
60  20  40  3
40  20  20  2
95  19  76  5
76  19  57  4
57  19  38  3
38  19  19  2
90  18  72  5
72  18  54  4
54  18  36  3
36  18  18  2
85  17  68  5
68  17  51  4
51  17  34  3
34  17  17  2
96  16  80  6
80  16  64  5
64  16  48  4
48  16  32  3
32  16  16  2
90  15  75  6
75  15  60  5
60  15  45  4
45  15  30  3
30  15  15  2
98  14  84  7
84  14  70  6
70  14  56  5
56  14  42  4
42  14  28  3
28  14  14  2
91  13  78  7
78  13  65  6
65  13  52  5
52  13  39  4
39  13  26  3
26  13  13  2
96  12  84  8
84  12  72  7
72  12  60  6
60  12  48  5
48  12  36  4
36  12  24  3
24  12  12  2
99  11  88  9
88  11  77  8
77  11  66  7
66  11  55  6
55  11  44  5
44  11  33  4
33  11  22  3
22  11  11  2
100 10  90  10
90  10  80  9
80  10  70  8
70  10  60  7
60  10  50  6
50  10  40  5
40  10  30  4
30  10  20  3
20  10  10  2
99  9   90  11
90  9   81  10
81  9   72  9
72  9   63  8
63  9   54  7
54  9   45  6
45  9   36  5
36  9   27  4
27  9   18  3
18  9   9   2
96  8   88  12
88  8   80  11
80  8   72  10
72  8   64  9
64  8   56  8
56  8   48  7
48  8   40  6
40  8   32  5
32  8   24  4
24  8   16  3
16  8   8   2
98  7   91  14
91  7   84  13
84  7   77  12
77  7   70  11
70  7   63  10
63  7   56  9
56  7   49  8
49  7   42  7
42  7   35  6
35  7   28  5
28  7   21  4
21  7   14  3
14  7   7   2
96  6   90  16
90  6   84  15
84  6   78  14
78  6   72  13
72  6   66  12
66  6   60  11
60  6   54  10
54  6   48  9
48  6   42  8
42  6   36  7
36  6   30  6
30  6   24  5
24  6   18  4
18  6   12  3
12  6   6   2
100 5   95  20
95  5   90  19
90  5   85  18
85  5   80  17
80  5   75  16
75  5   70  15
70  5   65  14
65  5   60  13
60  5   55  12
55  5   50  11
50  5   45  10
45  5   40  9
40  5   35  8
35  5   30  7
30  5   25  6
25  5   20  5
20  5   15  4
15  5   10  3
10  5   5   2
100 4   96  25
96  4   92  24
92  4   88  23
88  4   84  22
84  4   80  21
80  4   76  20
76  4   72  19
72  4   68  18
68  4   64  17
64  4   60  16
60  4   56  15
56  4   52  14
52  4   48  13
48  4   44  12
44  4   40  11
40  4   36  10
36  4   32  9
32  4   28  8
28  4   24  7
24  4   20  6
20  4   16  5
16  4   12  4
12  4   8   3
8   4   4   2
99  3   96  33
96  3   93  32
93  3   90  31
90  3   87  30
87  3   84  29
84  3   81  28
81  3   78  27
78  3   75  26
75  3   72  25
72  3   69  24
69  3   66  23
66  3   63  22
63  3   60  21
60  3   57  20
57  3   54  19
54  3   51  18
51  3   48  17
48  3   45  16
45  3   42  15
42  3   39  14
39  3   36  13
36  3   33  12
33  3   30  11
30  3   27  10
27  3   24  9
24  3   21  8
21  3   18  7
18  3   15  6
15  3   12  5
12  3   9   4
9   3   6   3
6   3   3   2
100 2   98  50
98  2   96  49
96  2   94  48
94  2   92  47
92  2   90  46
90  2   88  45
88  2   86  44
86  2   84  43
84  2   82  42
82  2   80  41
80  2   78  40
78  2   76  39
76  2   74  38
74  2   72  37
72  2   70  36
70  2   68  35
68  2   66  34
66  2   64  33
64  2   62  32
62  2   60  31
60  2   58  30
58  2   56  29
56  2   54  28
54  2   52  27
52  2   50  26
50  2   48  25
48  2   46  24
46  2   44  23
44  2   42  22
42  2   40  21
40  2   38  20
38  2   36  19
36  2   34  18
34  2   32  17
32  2   30  16
30  2   28  15
28  2   26  14
26  2   24  13
24  2   22  12
22  2   20  11
20  2   18  10
18  2   16  9
16  2   14  8
14  2   12  7
12  2   10  6
10  2   8   5
8   2   6   4
6   2   4   3
4   2   2   2

Now I list all solutions that can be removed after step 1 because there is only a single entry with that specific d

x   y   d   r
100 2   98  50
100 5   95  20
96  2   94  48
96  3   93  32
98  7   91  14
90  3   87  30
88  2   86  44
90  5   85  18
84  2   82  42
76  2   74  38
94  47  47  2
86  43  43  2
82  41  41  2
74  37  37  2
62  31  31  2
58  29  29  2
46  23  23  2
38  19  19  2
34  17  17  2
26  13  13  2
22  11  11  2
14  7   7   2
10  5   5   2
6   3   3   2
4   2   2   2

Now I list all solutions that can be removed after step 3 because there is only a single entry with that specific r

x   y   d   r
98  2   96  49
94  2   92  47
92  2   90  46
90  2   88  45
86  2   84  43
82  2   80  41
80  2   78  40
78  2   76  39
74  2   72  37
72  2   70  36
70  2   68  35
68  2   66  34
64  2   62  32
60  2   58  30

Now I list all solutions after step 6 because there is only a single entry with that specific d

x   y   d   r
96  4   92  24
93  31  62  3
87  29  58  3

I guess I made a mistake somewhere since I have 2 different r values in there but I assume that the correct ones would be with r=3 because there are two of those

Answer 4

EDIT: This answer is wrong but is kept in case someone thinks the same!

I think this question is flawed. This is my thought process:

From statement 1 we only know that $d$ is not $98$ with $x = 100$ and $y = 2$ because that is the only $d$ that has one solution.

Now we need to find what are the possible values of $r$ which only has one solution. This is only the case when $r > 33$ and $r < 50$ because if $r <= 33$ then there are at least two solutions taking $y = 2$ or $y = 3$ because $3 * 34 > 100$.

So $r <= 33$ according to statement 3.

Statement 4 is true but doesn't make sense to say. This statement is something that Bob also knew because if Bob knows the solution in statement 3 than $y$ is always $2$ and $x = d + 2$. And he knows how Alice thinks.

Statement 5 also has no added value because it follows from statement 4.

Statement 6 is also logically true because no information is gained since statement 3

Statement 7 can't possibly be true then