Find two numbers based on their sum/difference

For each pair $(A,B)$ the sum-difference pair $(S,D)$ is unique, and we can easily recover the values $(A,B) = (\tfrac12(S+D),\tfrac12(S-D)$ . Therefore my analysis is based purely on the sums and differences.

In the table below, I plotted the sums (Logician 1) vertically, and the differences (Logician 2) horizontally. The crosses mark the 283 possible $(S,D)$ pairs.

Each of the statements gives more information, thereby ruling out certain values for sum or difference. I did this ruling out by coloring columns and rows in the table.

Red : sums with only one cross. These sums are ruled out by statement 1.

Orange : differences with some crosses in red. These differences are ruled out by statement 2.

Yellow : sums with some crosses in orange. These sums are ruled out by statement 3.

Green : differences with no crosses in yellow. These differences satisfy (erroneous) statement 4.

Purple : sums with no crosses in green. In these situations, Logician 1 should now have realized that there are no solutions left, and he would have complained.

Blue : sums with one cross in green. In this situation, Logician 1 would have thought (incorrectly) that he knew the numbers; therefore, ruled out by statement 5.

White : Now that Logician 2 has replaced statement 4 by its opposite, all green columns are ruled out.

Only one cross in white remains , namely at $S = 99$ and $D = 33$ . Therefore the solution is $A = 33$ and $B = 66$ .

Here is computer code that generates the solution automatically:

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final int maxN = 500, nS = 150, nD = 98;

int A[] = new int[maxN];
int B[] = new int[maxN];
int S[] = new int[maxN];
int D[] = new int[maxN];
int x[] = new int[maxN];
int y[] = new int[maxN];

// generate possible pairs

int N = 0;
for (int a = 2; a <= 50; a++) 
    for (int b = 2*a; b <= 100; b += a, N++) {
        A[N] = a; B[N] = b; S[N] = a+b; D[N] = b-a; x[N] = 0; y[N] = 0; 
}

// Logician 1: "I don't know them."
// If a sum occurs more than once, promote x from 0 to 1.

for (int s = 1; s <= nS; s++) {
    int ct = 0;
    for (int i = 0; i < N; i++) if (S[i] == s) ct++;
    if (ct > 1)
        for (int i = 0; i < N; i++) if (S[i] == s) x[i] = 1;
}

// Logician 2: "I already knew that."
// If a difference is entirely marked x = 1, promote y from 0 to 2.

for (int d = 1; d <= nD; d++) {
    boolean pass = true;
    for (int i = 0; i < N; i++) if (D[i] == d)
        if (x[i] < 1) pass = false;
    if (pass)
        for (int i = 0; i < N; i++) if (D[i] == d) y[i] = 2;
}

// Logician 1: "I already know that you are supposed to know that."
// If a sum is entirely marked y = 2, promote x from 1 to 3.

for (int s = 1; s <= nS; s++) {
    boolean pass = true;
    for (int i = 0; i < N; i++) if (S[i] == s)
        if (y[i] < 2) pass = false;
    if (pass)
        for (int i = 0; i < N; i++) if (S[i] == s) x[i] = 3;
}

// Logician 2: "I think that you were about to say that!"
// If a difference is entirely marked x = 3, promote y from 2 to 4.

for (int d = 1; d <= nD; d++) {
    boolean pass = true;
    for (int i = 0; i < N; i++) if (D[i] == d)
        if (x[i] < 3) pass = false;
    if (pass)
        for (int i = 0; i < N; i++) if (D[i] == d) y[i] = 4;
}


// Logician 1: "I still can't find out what the numbers are."
// If a sum has no y = 4, set x = 5. This means no possible solutions.
// If a sum has more than one y = 4, set x = 6.

for (int s = 1; s <= nS; s++) {
    int ct = 0;
    for (int i = 0; i < N; i++) if (S[i] == s)
        if (y[i] == 4) ct++;
    if (ct < 1)
    { for (int i = 0; i < N; i++) if (S[i] == s) x[i] = 5; }
    if (ct > 1)
    { for (int i = 0; i < N; i++) if (S[i] == s) x[i] = 6; }
}

// Logician 2: "Oops, my last statement was wrong."
// Thus we must find an item with x = 6 and y = 2.

for (int i = 0; i < N; i++) if (x[i] == 6 && y[i] == 2) {
    System.out.println(""+A[i]+"."+B[i]);
}

Output: 33.66

In the following numbers are represented ike a-b/a since a divides b, exemple: 33-2 represents 33.66

From the conversation that shows, we could extract two lists containing all combinations that had summer and subtractor in mind, demonstrating what is discarded from all available sums in which stage for which reason... The bounded sum/difference by two brackets ex:[number] are discarded ones, the reason follows in same line.

Sums

Differences

from both lists we could extract such a selection:

valid sums: - after third declaration

16 $\ \$ 2-7 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ 4-3 $\ \ \ \ \ \ \ \ \ \ \ \$

28 $\ \$ 2-13 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 4-6 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 7-3 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$

32 $\ \$ 2-15 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 4-7 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 8-3 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$

76 $\ \$ 2-37 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 4-18 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 19-3 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

78 $\ \$ 2-38 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 3-25 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 6-12 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 13-5 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ 26-2 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

88 $\ \$ 2-43 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 4-21 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 8-10 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 11-7 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ 22-3 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

15 $\ \$ 3-4 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ 5-2 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

27 $\ \$ 3-8 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ 9-2 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$

33 $\ \$ 3-10 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 11-2 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

39 $\ \$ 3-12 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 13-2 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

57 $\ \$ 3-18 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 19-2 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

69 $\ \$ 3-22 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 23-2 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

87 $\ \$ 3-28 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 29-2 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

93 $\ \$ 3-30 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 31-2 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

99 $\ \$ 3-32 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 9-10 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 11-8 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 33-2 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

35 $\ \$ 5-6 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ 7-4 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

95 $\ \$ 5-18 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 19-4 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

77 $\ \$ 7-10 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 11-6 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

91 $\ \$ 7-12 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 13-6 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$

112 $\ \$ 14-7 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 16-6 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 28-3 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$

114 $\ \$ 19-5 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 38-2 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$

120 $\ \$ 20-5 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 24-4 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 30-3 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 40-2 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$

132 $\ \$ 33-3 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 44-2 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$

valid differences: - after fourth declaration

74 $\ \ \ \$ 2-38 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ .

9 $\ \ \ \ \$ 3-4 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ 9-2 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ .

81 $\ \ \ \$ 3-28 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ 9-10 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ .

87 $\ \ \ \$ 3-30 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ .

93 $\ \ \ \$ 3-32 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ .

5 $\ \ \ \ \$ 5-2 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ .

85 $\ \ \ \$ 5-18 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ .

77 $\ \ \ \$ 7-12 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ 11-8 $\ \ \$ .

11 $\ \ \ \$ 11-2 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ .

13 $\ \ \ \$ 13-2 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ .

19 $\ \ \ \$ 19-2 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ .

23 $\ \ \ \$ 23-2 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ .

29 $\ \ \ \$ 29-2 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ .

31 $\ \ \ \$ 31-2

Now from both lists this is the figuration of patterns in summer's head, where each sum contains atleast one couple concerned from 4th declaration, that is hence tagged by a star. The starred sums are all where summer still doesnt know .

78 $\ \ \$ 2-38* $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 3-25 $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 6-12 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ 13-5 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ 26-2 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

15* $\ \$ 3-4* $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ 5-2* $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

27 $\ \ \$ 3-8 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ 9-2* $\ \ \ \ \ \ \ \ \ \ \ \ \ \$

33 $\ \ \$ 3-10 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ 11-2* $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

39 $\ \ \$ 3-12 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ 13-2* $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

57 $\ \ \$ 3-18 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ 19-2* $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

69 $\ \ \$ 3-22 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ 23-2* $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

87* $\ \$ 3-28* $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 29-2* $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

93* $\ \$ 3-30* $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 31-2* $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

99* $\ \$ 3-32* $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 9-10* $\ \ \ \ \ \ \ \ \ \ \ \ \$ 11-8* $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 33-2 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

95 $\ \ \$ 5-18* $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 19-4 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

91 $\ \ \$ 7-12* $\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 13-6

Now, since subtractor has made a slip of his tongue, no one from the starred lines contains more than one starred couple, and atleast one couple not signified, except the sum=99 where the last disclaimer declared by subtractor is enough (but not easy) for any arbitrary witness from the audience to precise the numbers, those are 33-2 (twisted format of 33.66)

Thank you for your attention.