COMMENT ⊗   VALID 00004 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00002 00002	1. The numbers have no prime factor greater than 50, since otherwise this
C00004 00003	New start:
C00005 00004	New start:
C00006 ENDMK
C⊗;
1. The numbers have no prime factor greater than 50, since otherwise this
factor must be one of the numbers, and P would know that.

2. The sum must be odd since every even number (in that size range anyway)
is the sum of two primes, so that S could not be sure that P didn't know
the numbers.  Not quite!! If (even S) ∧ S > 100, it might be that S is
not the sum of two primes less than 100.

3. Since the sum is odd, one of the numbers is even, and P knows that
2 is a factor.

4. S = 2+prime with prime < 100 is excluded, because
otherwise S could not be sure that P didn't know.

5. If there were 2 2s, then the information that S knew that P didn't know
would inform P that both 2s are together.

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47

exclusions
s = 49,

6. "I knew you don't know" excludes s = 4+p with p<100.  

7. 4 and 6 exclude S < 29.

8. odd S ⊃ S ≤ 53.  For if S = 53+2n and P = 53*2*n, then P will know that
one of the numbers is 53.
New start:

1. Suppose S = prime + 2↑k.  For all S knows, it might be
m = prime and n = 2↑k.  In that case P = prime*2↑k, and after
S announced "I knew you don't know", P could reason as follows:
"If he knew I don't know, then he knew I didn't know.  Therefore,
S can't be even, since every even number is the sum of two primes,
so the only possibility is prime and 2↑k".

	This seems to eliminate any integer expressible in this way,
but this includes all odd numbers ≤ 55.

We seem to have reached a contradiction in our interpretation of
the problem.
New start:

Assume it reads "I knew you didn't know":

m + n ≤ 54 still follows: except for m + n = 197 or 198 which are
excluded by the fact that S doesn't know.

m + n odd still follows by Goldbach.

m+n ≠ prime + 2 still follows.

This leaves 11, 17, 23, 27, 29, 35, 37, 41, 47, 51, 53 as possibilities
for m + n.