Malmer Pebane, Fames Jung, and Weven Dare are perfect logicians that always tell the truth. Malmer decides to pose a puzzle to his friends: he tells them that the day of his birthday is at most the number of the month of his birthday. Then Malmer announces that he will whisper the day of his birthday to Fames and the month of his birthday to Weven, and he does exactly that.

After Malmer whispers to both of them, Fames thinks a bit, then says “Weven cannot know what Malmer’s birthday is.”

After that, Weven thinks a bit, then says “Fames also cannot know what Malmer’s birthday is.”

This exchange repeats, with Fames and Weven speaking alternately and each saying the other can’t know Malmer’s birthday. However, at one point, Weven instead announces “Fames and I can now know what Malmer’s birthday is. Interestingly, that was the longest conversation like that we could have possibly had before both figuring out Malmer’s birthday.”

Find Malmer’s birthday.
1. **Initial Setup and Constraints:**
   - Malmer tells Fames the day \( f \) and Weven the month \( w \).
   - Given \( f \leq w \), \( f \leq 31 \), and \( w \leq 12 \).

2. **First Statement Analysis:**
   - Fames says, "Weven cannot know what Malmer’s birthday is."
   - This implies that \( f \) is not uniquely identifying the month \( w \). If \( f \) were 1, Weven would know the birthday immediately since \( f \leq w \) would force \( w = 1 \). Therefore, \( f \neq 1 \).

3. **Second Statement Analysis:**
   - Weven says, "Fames also cannot know what Malmer’s birthday is."
   - This implies that \( w \) is not uniquely identifying the day \( f \). If \( w \) were 12, Fames would know the birthday immediately since \( f \leq w \) would force \( f = 12 \). Therefore, \( w \neq 12 \).

4. **Narrowing Down the Range:**
   - After the first exchange, we have \( 2 \leq f \leq 11 \) and \( 2 \leq w \leq 11 \).

5. **Subsequent Exchanges:**
   - Each subsequent exchange further narrows down the range of possible values for \( f \) and \( w \).
   - After the second exchange: \( 3 \leq f \leq 10 \) and \( 3 \leq w \leq 10 \).
   - After the third exchange: \( 4 \leq f \leq 9 \) and \( 4 \leq w \leq 9 \).
   - After the fourth exchange: \( 5 \leq f \leq 8 \) and \( 5 \leq w \leq 8 \).
   - After the fifth exchange: \( 6 \leq f \leq 7 \) and \( 6 \leq w \leq 7 \).

6. **Final Exchange:**
   - At this point, Fames says, "Weven cannot know the birthday."
   - This implies \( f = 7 \) because if \( f = 6 \), Weven would know the birthday immediately since \( f \leq w \) would force \( w = 6 \).
   - Therefore, \( f = 7 \) and \( w = 7 \).

7. **Conclusion:**
   - Weven then announces that both of them know the birthday, as \( f = 7 \) and \( w = 7 \) uniquely identify the birthday.

The final answer is \(\boxed{\text{July } 7^\text{th}}\).