Impossible Puzzle - Detailed Solution

Detailed Solution

Mathematician P

P knows p=52. P suspects (2,26) and (4,13). P knows s=28 or s=17.

If s=28:

S would suspect (2,26), (3,25), (4,24), (5,23), (6,22), (7,21), (8,20), (9,19), (10,18), (11,17), (12,16), and (13,15).

S would know if (5,23) or (11,17), P would know the numbers.

S would not be able to say "I was sure that you could not find them."

If s=17:

S would suspect (2,15), (3,14), (4,13), (5,12), (6,11), (7,10), and (8,9).

S would know that P would not know the numbers.

S would be able to say "I was sure that you could not find them."

Therefore, when S says "I was sure that you could not find them," P eliminates (2,26) and learns (4,13) is the answer.

Mathematician S

S knows s=17. S suspects (2,15), (3,14), (4,13), (5,12), (6,11), (7,10), and (8,9). S knows p is 30, 42, 52, 60, 66, 70, or 72.

When P says "Then, I found these numbers," S knows his statement eliminated all but one possibility for P.

S simulates P's thinking

Case 1 (p=30)

P knows p=30. P suspects (2,15), (3,10), and (5,6). P knows s is 17, 13, or 11.

If s=17:

S would suspect (2,15), (3,14), (4,13), (5,12), (6,11), (7,10), and (8,9).

S would know that P would not know the numbers.

S would be able to say "I was sure that you could not find them."

If s=13:

S would suspect (2,11), (3,10), (4,9), (5,8), and (6,7).

S would know if (2,11), P would know the numbers.

S would not be able to say "I was sure that you could not find them."

If s=11:

S would suspect (2,9), (3,8), (4,7), and (5,6).

S would know that P would not know the numbers.

S would be able to say "I was sure that you could not find them."

Therefore, when S says "I was sure that you could not find them," P eliminates (3,10) but cannot decide between (2,15) and (5,6).

Case 2 (p=42)

P knows p=42. P suspects (2,21), (3,14), and (6,7). P knows s is 23, 17, or 13.

If s=23:

S would suspect (2,21), (3,20), (4,19), (5,18), (6,17), (7,16), (8,15), (9,14), (10,13), and (11,12).

S would know that P would not know the numbers.

S would be able to say "I was sure that you could not find them."

If s=17:

S would suspect (2,15), (3,14), (4,13), (5,12), (6,11), (7,10), and (8,9).

S would know that P would not know the numbers.

S would be able to say "I was sure that you could not find them."

If s=13:

S would suspect (2,11), (3,10), (4,9), (5,8), and (6,7).

S would know if (2,11), P would know the numbers.

S would not be able to say "I was sure that you could not find them."

Therefore, when S says "I was sure that you could not find them," P eliminates (6,7) but cannot decide between (2,21) and (3,14).

Case 3 (p=52)

See Mathematician P's actual reasoning above.

Case 4 (p=60)

P knows p=60. P suspects (2,30), (3,20), (4,15), (5,12), and (6,10). P knows s is 32, 23, 19, 17, or 16.

If s=32:

S would suspect (2,30), (3,29), (4,28), (5,27), (6,26), (7,25), (8,24), (9,23), (10,22), (11,21), (12,20), (13,19), (14,18), and (15,17).

S would know if (3,29) or (13,19), P would know the numbers.

S would not be able to say "I was sure that you could not find them."

If s=23:

S would suspect (2,21), (3,20), (4,19), (5,18), (6,17), (7,16), (8,15), (9,14), (10,13), and (11,12).

S would know that P would not know the numbers.

S would be able to say "I was sure that you could not find them."

If s=19:

S would suspect (2,17), (3,16), (4,15), (5,14), (6,13), (7,12), (8,11), and (9,10).

S would know if (2,17), P would know the numbers.

S would not be able to say "I was sure that you could not find them."

If s=17:

S would suspect (2,15), (3,14), (4,13), (5,12), (6,11), (7,10), and (8,9).

S would know that P would not know the numbers.

S would be able to say "I was sure that you could not find them."

If s=16:

S would suspect (2,14), (3,13), (4,12), (5,11), (6,10), and (7,9).

S would know if (3,13) or (5,11), P would know the numbers.

S would not be able to say "I was sure that you could not find them."

Therefore, when S says "I was sure that you could not find them," P eliminates (2,30), (4,15), and (6,10) but cannot decide between (3,20) and (5,12).

Case 5 (p=66)

P knows p=66. P suspects (2,33), (3,22), and (6,11). P knows s is 35, 25, or 17.

If s=35:

S would suspect (2,33), (3,32), (4,31), (5,30), (6,29), (7,28), (8,27), (9,26), (10,25), (11,24), (12,23), (13,22), (14,21), (15,20), (16,19), and (17,18).

S would know that P would not know the numbers.

S would be able to say "I was sure that you could not find them."

If s=25:

S would suspect (2,23), (3,22), (4,21), (5,20), (6,19), (7,18), (8,17), (9,16), (10,15), (11,14), and (12,13).

S would know if (2,23), P would know the numbers.

S would not be able to say "I was sure that you could not find them."

If s=17:

S would suspect (2,15), (3,14), (4,13), (5,12), (6,11), (7,10), and (8,9).

S would know that P would not know the numbers.

S would be able to say "I was sure that you could not find them."

Therefore, when S says "I was sure that you could not find them," P eliminates (3,22) but cannot decide between (2,33) and (6,11).

Case 6 (p=70)

P knows p=70. P suspects (2,35), (5,14), and (7,10). P knows s is 37, 19, or 17.

If s=37:

S would suspect (2,35), (3,34), (4,33), (5,32), (6,31), (7,30), (8,29), (9,28), (10,27), (11,26), (12,25), (13,24), (14,23), (15,22), (16,21), (17,20), and (18,19).

S would know that P would not know the numbers.

S would be able to say "I was sure that you could not find them."

If s=19:

S would suspect (2,17), (3,16), (4,15), (5,14), (6,13), (7,12), (8,11), and (9,10).

S would know if (2,17), P would know the numbers.

S would not be able to say "I was sure that you could not find them."

If s=17:

S would suspect (2,15), (3,14), (4,13), (5,12), (6,11), (7,10), and (8,9).

S would know that P would not know the numbers.

S would be able to say "I was sure that you could not find them."

Therefore, when S says "I was sure that you could not find them," P eliminates (5,14) but cannot decide between (2,35) and (7,10).

Case 7 (p=72)

P knows p=72. P suspects (2,36), (3,24), (4,18), (6,12), and (8,9). P knows s is 38, 27, 22, 18, or 17.

If s=38:

S would suspect (2,36), (3,35), (4,34), (5,33), (6,32), (7,31), (8,30), (9,29), (10,28), (11,27), (12,26), (13,25), (14,24), (15,23), (16,22), (17,21), and (18,20).

S would know that if (7,31), P would know the numbers.

S would not be able to say "I was sure that you could not find them."

If s=27:

S would suspect (2,25), (3,24), (4,23), (5,22), (6,21), (7,20), (8,19), (9,18), (10,17), (11,16), (12,15), and (13,14).

S would know that P would not know the numbers.

S would be able to say "I was sure that you could not find them."

If s=22:

S would suspect (2,20), (3,19), (4,18), (5,17), (6,16), (7,15), (8,14), (9,13), and (10,12).

S would know if (3,19) or (5,17), P would know the numbers.

S would not be able to say "I was sure that you could not find them."

If s=18:

S would suspect (2,16), (3,15), (4,14), (5,13), (6,12), (7,11), and (8,10).

S would know if (5,13) or (7,11), P would know the numbers.

S would not be able to say "I was sure that you could not find them."

If s=17:

S would suspect (2,15), (3,14), (4,13), (5,12), (6,11), (7,10), and (8,9).

S would know that P would not know the numbers.

S would be able to say "I was sure that you could not find them."

Therefore, when S says "I was sure that you could not find them," P eliminates (2,36), (4,18), and (6,12) but cannot decide between (3,24) and (8,9).

Only Case 3 eliminates all but one possibility for P. That's how S decides (4,13) is the answer.