I’ll try again.
Let’s use the following terminology:
r = a man with two red stamps; b = man with two blue stamps; x = man with a mix.
x' = NOT (x was unable to deduce his stamps.)
* = AND;
+ = OR;
x(condition) = x can deduce his stamps based upon the condition. E.g., x(r’) means x can deduce his stamps after r has failed.
There are only 6 significant cases (disregarding the order) based on the number of mixed colors in the lineup -- from zero to 5: rrrbb, xrrbb, xxrrb, xxxrb, xxxxr, xxxxx. “r” and “b” may be swapped for one another making logically identical cases.
The 6 cases may be resolved as following:
1) r1 r2 r3 b1 b2 ==> b() + r3(r2’(r1’)).
Example 1: r1’ b1 r2 r3 b2, where b1 solves on his turn; OR Ex.2: r1’ r2’ r3 b1 b2, where r3 solves after r1 and r2 failed.
2) x r1 r2 b1 b2==> x(r’ * b’) + r2(r1’) + b2(b1’)
Ex.1: r1’ b1’ x r2 b2. Ex.2: r1’ r2 b1 b2 x. Ex.3: b1’ x’ r1’ b2 r2.
3) x1 x2 r1 r2 b ==> b() + x(r2’(r1’)).
Ex.1: b r1 r2 x1 x2. Ex.2: r1’ r2’ x1 b x2.
4) x1 x2 x3 r b ==> x(r’ * b’).
Example: x1’ r’ b’ x2 x3.
5) x1 x2 x3 x4 r ==> x1(x2’(x1’ * r’)).
Ex.1: x1' r' x2' x3' x4' (x1 solves on his second turn). Ex.2: x1'' x2' x3' x4' r' (x2 solves on his second turn.)
6) x1 x2 x3 x4 x5 ==> x2(x1’(x’(x1’ * x2’))).
Example: x1'' x2' x3' x4' x5', where x2 can deduce that he is a mix on his second turn after x1 fails for the second time.
I derive deduction strings for each case based on perspectives that the men in the lineup may have and deduction strings from those perspectives. Take for example case (3). An x in that line up may suppose he is either part of the case (2), or (3). Without any additional data, r may think he is in either (3) or (4). Whereas b in lineup (3) simply knows he is b.
It appears, the longest deduction string here is one where the men made one full round of non-guessing whereafter in the second round the second man in the line up must deduce his stamps. There are several possibilities for that:
x1'' x2' x3' x4' x5';
x1'' x2' x3' x4' r';
r'' x1' x2' x3' b'.
The first round, where 14 stamps were in play, does not seem to offer any useful information for those longest deduction variations above.
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