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If one of them has a number greater than 8 he/she would know that the sum is 15 as it can't be 5 or 8 anymore.

 

For the other numbers the following table is true(after both said no):

 

own  other
0    5, 8 -> no 15 as the other person would have known it
1    4, 7 -> no 14 as the other person would have known it
2    3, 6 -> no 13 as the other person would have known it
3    2, 5 -> no 12 as the other person would have known it
4    1, 4 -> no 11 as the other person would have known it
5    0, 3 -> no 10 as the other person would have known it
6    2    -> no 9 as the other person would have known it
7    1, 8 
8    0, 7

If one of them has a 6 he/she now knows the number of the other one

 

For the other numbers the following table is true(after both said no):

 

own  other
0    5, 8
1    4, 7
2    3    -> no 6 as the other person would have known it
3    2, 5
4    1, 4
5    0, 3
7    1, 8 
8    0, 7

If one of them has a 2 he/she now knows the number of the other one

 

For the other numbers the following table is true(after both said no):

 

own  other
0    5, 8
1    4, 7
3    5    -> no 2 as the other person would have known it
4    1, 4
5    0, 3
7    1, 8 
8    0, 7

If one of them has a 3 he/she now knows the number of the other one

 

For the other numbers the following table is true(after both said no):

 

own  other
0    5, 8
1    4, 7
4    1, 4
5    0    -> no 3 as the other person would have known it
7    1, 8 
8    0, 7

If one of them has a 5 he/she now knows the number of the other one

 

For the other numbers the following table is true(after both said no):

 

own  other
0    8    -> no 5 as the other person would have known it
1    4, 7
4    1, 4
7    1, 8 
8    0, 7

If one of them has a 0 he/she now knows the number of the other one

 

For the other numbers the following table is true(after both said no):

 

own  other
1    4, 7
4    1, 4
7    1, 8 
8    7    -> no 0 as the other person would have known it

If one of them has a 8 he/she now knows the number of the other one

 

For the other numbers the following table is true(after both said no):

 

own  other
1    4, 7
4    1, 4
7    1    -> no 8 as the other person would have known it

If one of them has a 7 he/she now knows the number of the other one

 

For the other numbers the following table is true(after both said no):

 

own  other
1    4    -> no 7 as the other person would have known it
4    1, 4

If one of them has a 1 he/she now knows the number of the other one

 

For the other numbers the following table is true(after both said no):

 

own  other
4    4    -> no 1 as the other person would have known it

If one of them has a number greater than 8 he/she would know that the sum is 15 as it can't be 5 or 8 anymore.

For the other numbers the following table is true(after both said no):

own  other
0    5, 8 -> no 15 as the other person would have known it
1    4, 7 -> no 14 as the other person would have known it
2    3, 6 -> no 13 as the other person would have known it
3    2, 5 -> no 12 as the other person would have known it
4    1, 4 -> no 11 as the other person would have known it
5    0, 3 -> no 10 as the other person would have known it
6    2    -> no 9 as the other person would have known it
7    1, 8 
8    0, 7

If one of them has a 6 he/she now knows the number of the other one

For the other numbers the following table is true(after both said no):

own  other
0    5, 8
1    4, 7
2    3    -> no 6 as the other person would have known it
3    2, 5
4    1, 4
5    0, 3
7    1, 8 
8    0, 7

If one of them has a 2 he/she now knows the number of the other one

For the other numbers the following table is true(after both said no):

own  other
0    5, 8
1    4, 7
3    5    -> no 2 as the other person would have known it
4    1, 4
5    0, 3
7    1, 8 
8    0, 7

If one of them has a 3 he/she now knows the number of the other one

For the other numbers the following table is true(after both said no):

own  other
0    5, 8
1    4, 7
4    1, 4
5    0    -> no 3 as the other person would have known it
7    1, 8 
8    0, 7

If one of them has a 5 he/she now knows the number of the other one

For the other numbers the following table is true(after both said no):

own  other
0    8    -> no 5 as the other person would have known it
1    4, 7
4    1, 4
7    1, 8 
8    0, 7

If one of them has a 0 he/she now knows the number of the other one

For the other numbers the following table is true(after both said no):

own  other
1    4, 7
4    1, 4
7    1, 8 
8    7    -> no 0 as the other person would have known it

If one of them has a 8 he/she now knows the number of the other one

For the other numbers the following table is true(after both said no):

own  other
1    4, 7
4    1, 4
7    1    -> no 8 as the other person would have known it

If one of them has a 7 he/she now knows the number of the other one

For the other numbers the following table is true(after both said no):

own  other
1    4    -> no 7 as the other person would have known it
4    1, 4

If one of them has a 1 he/she now knows the number of the other one

For the other numbers the following table is true(after both said no):

own  other
4    4    -> no 1 as the other person would have known it

In the 100 blue eyes problem - why is the oracle necessary?

In the 100 blue eyes problem - why is the oracle necessary?

EDIT:

This is actually a kind of derivation of the blue eyes problem:

In the 100 blue eyes problem - why is the oracle necessary?