Answer: The third party (blind) concludes that his hat is white using second-order reasoning about silence.
1) Common information
- There are 5 hats available: 3 white and 2 black.
- 3 hats are used, one per wise man.
- The first two see two white hats (in the other two wise men).
- The third does not see anything, but hears perfectly rational responses.
2) First response: wise man 1 says "I don't know"
That, in itself, is compatible with the third party being black or white.
There is still no conclusion for the blind wise man.
3) Key: second answer "I don't know" from wise man 2
The blind wise man analyzes by contradiction:
- Hypothesis H: "my hat is black."
- Under H, wise man 2 sees: wise man 1 with white and wise man 3 (blind) with black.
- So, upon hearing that wise man 1 said "I don't know", wise man 2 can reason:
- "If I had black, wise man 1 would see two black people (mine and the blind man's), and since there are only 2 black people in total, wise man 1 would immediately know that his is white."
- "But wise man 1 said 'I don't know', so I can't have black."
- "Therefore, I must have white."- That is, under H, the wise man 2 could know his color and would not say "I don't know".
Contradiction with what was observed (wise man 2 did say "I don't know").
Therefore H is false.
4) Conclusion
The blind wise man cannot have black, so his hat must necessarily be:
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\boxed{\text{white}}.
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