Okay, in R2, Suteguma knows "Baku's beads are 4-10", he knows that Baku knows "My beads are 1-9", and he thinks "Baku has fallen for my trick. He thinks my beads are really 1-7".

1. Because Suteguma isn't allowed to guess 1-9 in the Dotty (it would be an illegal guess), and isn't allowed to guess 10-12 (He and Baku already guessed those numbers). He's allowed to guess 13-19 (and if he guesses 18 or 19, he's giving up on tricking Baku, so he won't do that).

2. That means realistically, the numbers he can guess are 13, 14, 15, 16, or 17

3. Because the smallest number he can guess is 13, no matter what, Baku is going to know "Suteguma's beads are not 1 or 2." So Suteguma thinks the options available at the end of the round will be, at best, "He thinks my beads are 3-7".

4. Suteguma knows Baku doesn't have 1-3 beads (But Baku doesn't know Suteguma knows this). This is because Suteguma really has 9 beads, and if Baku had 1/2/3 beads, the game would have ended when they guessed 10/11/12 in Dotty earlier.

That's all confirmed. The rest is a bit of guesswork.

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If Suteguma guesses 17 in R2:

1. Baku will instantly know he doesn't have 1-6 beads, because that would be an illegal guess. The only option is Suteguma MUST have 7-9 beads (He can't have 10, because that would make Suteguma's guess of 10 in Dotty in R1 illegal).

2. If Baku's fallen for Suteguma's trick, he will think it's impossible for Suteguma to have 8-10 beads... but that's very suspicious. Both players should know this, so both players will know this guess is telling Baku "I definitely have 7 beads".

   2a. If Baku has 4/5/10 beads, he will instantly know this is a trick.

      i) If Baku has 4 beads, the only possible numbers left for Suteguma to possibly have are 3/4/5/9 beads (If Suteguma had 6/7/8 beads, the game would have ended earlier when they guessed 10/11/12 in Dotty). 17 would be an illegal guess if Suteguma has 3/4/5 beads. So Baku would immediately know "Suteguma has 9 beads" and win.

      ii) If Baku has 5 beads, the only possible numbers left for Suteguma to possibly have are 3/4/8/9 beads (If Suteguma had 5/6/7 beads, the game would have ended earlier when they guessed 10/11/12 in Dotty). 17 would be an illegal guess if Suteguma has 3/4 beads. So Baku would immediately know "Suteguma has 8 or 9 beads". Suteguma's trick would be ruined, and Baku would get a lot of information.

      ii) If Baku has 10 beads, once Suteguma guesses 17, Baku will immediately know "Suteguma must have 7-9 beads, or that would be an illegal guess. If he had 7 beads, he would have won just now... So he must have 8 or 9 beads." Suteguma's trick would be ruined, and Baku would get a lot of information.

   2b. Baku doesn't have 1-3 beads, guaranteed. If he doesn't have 4/5/10 beads, the only options left are 6-9 beads. If Baku has 6-9 beads, he will STILL probably suspect this is a trick - it's too suspicious to guess 17 here.

3. So no matter what, 17 is a bad guess for Suteguma. He won't do that.

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If Suteguma guesses 16 in R2:

1. Baku will instantly know he doesn't have 1-5 beads, because that would be an illegal guess. The only option is Suteguma MUST have 6-9 beads (He can't have 10, because that would make Suteguma's guess of 10 in Dotty in R1 illegal).

2. If Baku's fallen for Suteguma's trick, he will think it's impossible for Suteguma to have 8-10 beads. That means this trick is saying "I have either 6 or 7 beads".

   2a. If Baku has 4/5 beads, he will instantly know this is a trick.

      i) If Baku has 4 beads, the only possible numbers left for Suteguma to possibly have are 3/4/5/9 beads (If Suteguma had 6/7/8 beads, the game would have ended earlier when they guessed 10/11/12 in Dotty). 16 would be an illegal guess if Suteguma has 3/4/5 beads. So Baku would immediately know "Suteguma has 9 beads" and win.

      ii) If Baku has 5 beads, the only possible numbers left for Suteguma to possibly have are 3/4/8/9 beads (If Suteguma had 5/6/7 beads, the game would have ended earlier when they guessed 10/11/12 in Dotty). 16 would be an illegal guess if Suteguma has 3/4 beads. So Baku would immediately know "Suteguma has 8 or 9 beads". Suteguma's trick would be ruined, and Baku would get a lot of information.

   2b. Baku doesn't have 1-3 beads, guaranteed. If he doesn't have 4/5 beads, the only options left are 6-10 beads.

      i) If Baku has 6 beads, the only possible numbers left for Suteguma to possibly have are 7/8/9 beads. If Baku's fallen for Suteguma's trick and doesn't get suspicious, he might believe "Suteguma has 7 beads". That would likely let Suteguma win, if Baku doesn't get suspicious.

      ii) If Baku has 7 beads, Suteguma instantly wins because 7 + 9 = 16. No mind games are needed at this point.

      iii) If Baku has 8 beads, the only possible numbers left for Suteguma to possibly have are 6/7/9 beads (He can't have 8 beads, because if he did, the round would instantly end: 8 + 8 = 16). If Baku's fallen for Suteguma's trick and doesn't get suspicious, he might believe "Suteguma has either 6 beads or 7 beads". If he does get suspicious, he would instantly know the answer is 9.

      iv) If Baku has 9 beads, the only possible numbers left for Suteguma to possibly have are 6/9 beads (He can't have 7 beads, because if he did, the round would instantly end: 9 + 7 = 16). If Baku's fallen for Suteguma's trick and doesn't get suspicious, he might believe "Suteguma has 6 beads". That would likely let Suteguma win, if Baku doesn't get suspicious. If he does get suspicious, he would instantly know the answer is 9.

      v) If Baku has 10 beads, the only possible numbers left for Suteguma to possibly have are 7/8/9 beads. If Baku's fallen for Suteguma's trick and doesn't get suspicious, he might believe "Suteguma has 7 beads". That would likely let Suteguma win, if Baku doesn't get suspicious.

3. So 16 is a risky play.
It has a 1/7 chance of instantly winning, a 1/7 chance of instantly losing, a 1/7 chance of exposing his trick to Baku, a 3/7 chance of winning if Baku doesn't get suspicious at all, and a 1/7 chance of no major result unless Baku gets suspicious.

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If Suteguma guesses 15 in R2:

1. Baku will instantly know he doesn't have 1-4 beads, because that would be an illegal guess. The only option is Suteguma MUST have 5-9 beads (He can't have 10, because that would make Suteguma's guess of 10 in Dotty in R1 illegal).

2. If Baku's fallen for Suteguma's trick, he will think it's impossible for Suteguma to have 8-10 beads. That means this trick is saying "I have either 5, 6, or 7 beads".

   2a. If Baku has 5 beads, he will instantly know this is a trick.

      i) If Baku has 5 beads, the only possible numbers left for Suteguma to possibly have are 8/9 beads (If Suteguma had 5/6/7 beads, the game would have ended earlier when they guessed 10/11/12 in Dotty). So Baku would immediately know "Suteguma has 8 or 9 beads". Suteguma's trick would be ruined, and Baku would get a lot of information.

   2b. Baku doesn't have 1-3 beads, guaranteed. If he doesn't have 5 beads, the only options left are 4/6-10 beads.

      i) If Baku has 4 beads, the only possible numbers left for Suteguma to possibly have are 5/9 beads (If Suteguma had 6/7/8 beads, the game would have ended earlier when they guessed 10/11/12 in Dotty). If Baku's fallen for Suteguma's trick and doesn't get suspicious, he might believe "Suteguma has 5 beads". That would likely let Suteguma win, if Baku doesn't get suspicious.

      ii) If Baku has 6 beads, Suteguma instantly wins because 6 + 9 = 15. No mind games are needed at this point.

      iii) If Baku has 7 beads, the only possible numbers left for Suteguma to possibly have are 6/7/9 beads (If Suteguma had 5 beads, the game would have ended earlier when they guessed 12 in Dotty, and if Suteguma had 8 beads, the round would instantly end now because 7+8=15). If Baku's fallen for Suteguma's trick and doesn't get suspicious, he might believe "Suteguma has 6 or 7 beads". If he does get suspicious, he would instantly know the answer is 9.

      iv) If Baku has 8 beads, the only possible numbers left for Suteguma to possibly have are 5/6/8/9 beads (If Suteguma had 7 beads, the round would instantly end now because 8+7=15). If Baku's fallen for Suteguma's trick and doesn't get suspicious, he might believe "Suteguma has either 6 beads or 7 beads".

      v) If Baku has 9 beads, the only possible numbers left for Suteguma to possibly have are 5/7/8/9 beads (If Suteguma had 6 beads, the round would instantly end now because 9+6=15). If Baku's fallen for Suteguma's trick and doesn't get suspicious, he might believe "Suteguma has either 5 beads or 7 beads".

      vi) If Baku has 10 beads, the only possible numbers left for Suteguma to possibly have are 6/7/8/9 beads (If Suteguma had 5 beads, the round would instantly end now because 10+5=15). If Baku's fallen for Suteguma's trick and doesn't get suspicious, he might believe "Suteguma has either 5, 6, or 7 beads".

3. So 15 is an even riskier play!
It has a 1/7 chance of instantly winning, a 1/7 chance of instantly losing, a 2/7 chance of winning if Baku doesn't get suspicious at all, and a 3/7 chance of no major result unless Baku gets suspicious.

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If Suteguma guesses 14 in R2:

1. Baku will instantly know he doesn't have 1-3 beads, because that would be an illegal guess. The only option is Suteguma MUST have 4-9 beads (He can't have 10, because that would make Suteguma's guess of 10 in Dotty in R1 illegal).

2. If Baku's fallen for Suteguma's trick, he will think it's impossible for Suteguma to have 8-10 beads. That means this trick is saying "I have either 4, 5, 6, or 7 beads".

   2a. No matter how many beads he has, Baku will never have guaranteed proof that Suteguma's 8 was tricking him.

   2b. Baku doesn't have 1-3 beads, guaranteed. The only options left are 4-10 beads.

      i) If Baku has 4 beads, the only possible numbers left for Suteguma to possibly have are 4/5/9 beads (If Suteguma had 6/7/8 beads, the game would have ended earlier when they guessed 10/11/12 in Dotty). If Baku's fallen for Suteguma's trick and doesn't get suspicious, he might believe "Suteguma has 4 beads or 5 beads".

      ii) If Baku has 5 beads, Suteguma instantly wins because 5 + 9 = 15. No mind games are needed at this point.

      iii) If Baku has 6 beads, the only possible numbers left for Suteguma to possibly have are 7/9 beads (If Suteguma had 4/5/6 beads, the game would have ended earlier when they guessed 10/11/12 in Dotty, and if Suteguma had 8 beads, the round would instantly end now because 6+8=14). If Baku's fallen for Suteguma's trick and doesn't get suspicious, he might believe "Suteguma has 7 beads". That would likely let Suteguma win, if Baku doesn't get suspicious. If he does get suspicious, he would instantly know the answer is 9.

      iv) If Baku has 7 beads, the only possible numbers left for Suteguma to possibly have are 6/8/9 beads (If Suteguma had 4/5 beads, the game would have ended earlier when they guessed 11/12 in Dotty, and if Suteguma had 7 beads, the round would instantly end now because 7+7=14). If Baku's fallen for Suteguma's trick and doesn't get suspicious, he might believe "Suteguma has 6 beads". That would likely let Suteguma win, if Baku doesn't get suspicious.

      v) If Baku has 8 beads, the only possible numbers left for Suteguma to possibly have are 5/7/8/9 beads (If Suteguma had 4 beads, the game would have ended earlier when they guessed 12 in Dotty, and if Suteguma had 6 beads, the round would instantly end now because 8+6=14). If Baku's fallen for Suteguma's trick and doesn't get suspicious, he might believe "Suteguma has 5 or 7 beads".

      vi) If Baku has 9 beads, the only possible numbers left for Suteguma to possibly have are 4/6/7/8/9 beads (If Suteguma had 5 beads, the round would instantly end now because 9+5=14). If Baku's fallen for Suteguma's trick and doesn't get suspicious, he might believe "Suteguma has either 4, 6, or 7 beads".

      vii) If Baku has 10 beads, the only possible numbers left for Suteguma to possibly have are 5/6/7/8/9 beads (If Suteguma had 4 beads, the round would instantly end now because 10+4=14). If Baku's fallen for Suteguma's trick and doesn't get suspicious, he might believe "Suteguma has either 5, 6, or 7 beads".

3. So 14 is a middle-of-the-road play.
1/7 chance of instantly winning, a 1/7 chance of winning if Baku doesn't get suspicious at all, a 5/7 chance of no major result unless Baku gets suspicious, and no chance of giving away your trick.

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If Suteguma guesses 13 in R2:

1. Baku will instantly know he doesn't have 1-2 beads, because that would be an illegal guess. The only option is Suteguma MUST have 3-9 beads (He can't have 10, because that would make Suteguma's guess of 10 in Dotty in R1 illegal).

2. If Baku's fallen for Suteguma's trick, he will think it's impossible for Suteguma to have 8-10 beads. That means this trick is saying "I have either 3, 4, 5, 6, or 7 beads".

   2a. No matter how many beads he has, Baku will never have guaranteed proof that Suteguma's 8 was tricking him.

   2b. Baku doesn't have 1-3 beads, guaranteed. The only options left are 4-10 beads.

      i) If Baku has 4 beads, Suteguma instantly wins because 4 + 9 = 13. No mind games are needed at this point.

      ii) If Baku has 5 beads, the only possible numbers left for Suteguma to possibly have are 3/4/9 beads (If Suteguma had 5/6/7 beads, the game would have ended earlier when they guessed 10/11/12 in Dotty, and if Suteguma had 8 beads, the round would instantly end now because 5+8=13). If Baku's fallen for Suteguma's trick and doesn't get suspicious, he might believe "Suteguma has 6 or 7 beads". If he does get suspicious, he would instantly know the answer is 9.

      iii) If Baku has 6 beads, the only possible numbers left for Suteguma to possibly have are 3/8/9 beads (If Suteguma had 4/5/6 beads, the game would have ended earlier when they guessed 10/11/12 in Dotty, and if Suteguma had 7 beads, the round would instantly end now because 6+7=13). If Baku's fallen for Suteguma's trick and doesn't get suspicious, he might believe "Suteguma has 3 beads". That would likely let Suteguma win, if Baku doesn't get suspicious.

      iv) If Baku has 7 beads, the only possible numbers left for Suteguma to possibly have are 7/8/9 beads (If Suteguma had 3/4/5 beads, the game would have ended earlier when they guessed 10/11/12 in Dotty, and if Suteguma had 6 beads, the round would instantly end now because 7+6=13). If Baku's fallen for Suteguma's trick and doesn't get suspicious, he might believe "Suteguma has 7 beads". That would likely let Suteguma win, if Baku doesn't get suspicious.

      v) If Baku has 8 beads, the only possible numbers left for Suteguma to possibly have are 6/7/8/9 beads (If Suteguma had 3/4 beads, the game would have ended earlier when they guessed 11/12 in Dotty, and if Suteguma had 5 beads, the round would instantly end now because 8+5=13). If Baku's fallen for Suteguma's trick and doesn't get suspicious, he might believe "Suteguma has 6 or 7 beads".

      vi) If Baku has 9 beads, the only possible numbers left for Suteguma to possibly have are 5/6/7/8/9 beads (If Suteguma had 3 beads, the game would have ended earlier when they guessed 12 in Dotty, and if Suteguma had 4 beads, the round would instantly end now because 9+4=13). If Baku's fallen for Suteguma's trick and doesn't get suspicious, he might believe "Suteguma has either 5, 6, or 7 beads".

      vii) If Baku has 10 beads, the only possible numbers left for Suteguma to possibly have are 4/5/6/7/8/9 beads (If Suteguma had 3 beads, the round would instantly end now because 10+3=13). If Baku's fallen for Suteguma's trick and doesn't get suspicious, he might believe "Suteguma has either 4, 5, 6, or 7 beads".

3. So 13 is also a very middle-of-the-road play.
1/7 chance of instantly winning, a 2/7 chance of winning if Baku doesn't get suspicious at all, a 4/7 chance of no major result unless Baku gets suspicious, and no chance of giving away your trick.