Jin Huang smiled faintly. "You may begin!"
Lin Fan was surprised. "I should start first?"
Jin Huang nodded.
Current Game State: (12, 12, 12, 12, 12, 12, 12, 12, 12).
Lin Fan immediately did internal calculations. There are 5 possible moves - {1, 2, 3, 4, 6}.
Due to the symmetric nature of the cycle, no matter what pillar I choose, it's all the same. However, what number I should remove has to be calculated.
He also started thinking deeply about some of the properties of the game. What will happen if a pillar ring reduces to 1?
It will automatically lock the neighboring pillars.
For example, take Pillar 1. If Pillar 1 = 1, and the rest of the pillars are the same, in such a case, the common divisor corresponding to Pillar 2 and Pillar 9 would be the common divisor of Pillar 1 and Pillar 3, and the common divisor of Pillar 8 and Pillar 1, respectively. In both cases, the only common divisor is 1, and as a result, both the pillars are completely blocked.
Second thing to note is that, from any pillar, the maximum you can reduce is 11 rings. If a pillar has 12 rings, then the lowest number of rings it can actually have is 1. After you get to 1, there would be no more legal moves that can reduce it further, since the minimum number of rings you can remove is 1, and the maximum is half of the current ring count, rounded down to the closest whole number.
If the current state is 1: Minimum = 1, Maximum = 1/2 = 0 when rounded down. Minimum is greater than maximum, so you can't reduce further.
A pillar's strength is irrelevant if its neighbors are out of harmony. By intentionally manipulating two pillars so their ring counts share no common factors (making them co-prime, like 8 and 9, or 5 and 7), the pillar situated between them is instantly neutralized. The ultimate strategy is not to attack the pillar the opponent wants to use, but to subtly misalign its neighbors so the Greatest Common Divisor becomes 1, freezing it completely.
As Lin Fan continued thinking, Jin Huang smiled. "You have only 8 seconds left!"
Lin Fan kept thinking, and after a moment, he took a deep breath and said, "Pillar 1st, remove 2 rings!"
Board State After Move: (10, 12, 12, 12, 12, 12, 12, 12, 12).
Jin Huang nodded and immediately made her choice. "Pillar 3, remove 2 rings!"
Board State After Move: (10, 12, 10, 12, 12, 12, 12, 12, 12).
Lin Fan frowned. 'Targeting the Pillar 2nd?'
Lin Fan continued. "Pillar 1, remove 2 rings."
Board State After Move: (8, 12, 10, 12, 12, 12, 12, 12, 12).
In this way, the game continued. Jin Huang again targeted the Pillar 3rd.
Board State After Move: (8, 12, 9, 12, 12, 12, 12, 12, 12).
Round 3: Lin Fan's turn: (8, 12, 9, 12, 12, 12, 12, 12, 11), Jin Huang's turn: (8, 12, 9, 11, 12, 12, 12, 12, 11)
Round 4: Lin Fan's turn: (8, 12, 9, 11, 12, 12, 11, 12, 11), Jin Huang's turn: (8, 12, 9, 8, 12, 12, 11, 12, 11)
Round 5: Lin Fan's turn: (8, 12, 8, 8, 12, 12, 11, 12, 11), Jin Huang's turn: (8, 11, 8, 8, 12, 12, 11, 12, 11)
Round 6: Lin Fan's turn: (8, 11, 8, 6, 12, 12, 11, 12, 11), Jin Huang's turn: (8, 10, 8, 6, 12, 12, 11, 12, 11)
Round 7: Lin Fan's turn: (8, 10, 8, 6, 12, 12, 11, 11, 11), Jin Huang's turn: (8, 10, 8, 5, 12, 12, 11, 11, 11)
Round 8: Lin Fan's turn: (8, 10, 7, 5, 12, 12, 11, 11, 11).
At this moment, Jin Huang took 20 seconds to think. She suddenly looked up at Lin Fan and said, "I was waiting!"
Lin Fan was puzzled. "What do you mean?"
Jin Huang giggled. "I thought you had realized how much worse your position already is!"
Lin Fan's eyes narrowed as he heard her continue. "Right now, I have only two moves to make, and one of them leads to your victory, while the other leads to my victory! My victory is guaranteed with perfect play. If you want to make use of your advantage and make me choose a move that leads to my loss!"
Lin Fan asked subconsciously. "Can I get some time to think?"
Jin Huang nodded. "You have 60 seconds!"
Lin Fan immediately started thinking. 'There are only two possible choices - Pillar 3, reduce 1, or Pillar 8, reduce 1! Even if I make a choice, will I be able to convert it?'
He suddenly recalled what his master said. 'You need to impress her. The more brilliance you show, the more benefits you will obtain. However, you must play to win and never despair! Do not give up halfway!'
He started calculating a few moves ahead, even though he didn't know how important an impact they would make.
Choosing the 3rd pillar would give Jin Huang a lot of flexibility, and she would have 7 legal moves, while choosing the 8th pillar would give her only 6 legal moves.
He also tried to do a deeper search. What are the moves for each of the 13 legal moves? He calculated to a few depths, and then he started reaching his limits, as he heard a voice.
"Time is running out!"
Lin Fan shook his head and said, "I want you to choose Pillar 8!"
Jin Huang glanced at Lin Fan deeply and then nodded.
Board State After Move: (8, 10, 7, 5, 12, 12, 11, 10, 11).
Round 9: Lin Fan's turn: (8, 10, 6, 5, 12, 12, 11, 10, 11), Jin Huang's turn: (8, 9, 6, 5, 12, 12, 11, 10, 11)
Round 10: Lin Fan's turn: (8, 9, 6, 5, 12, 12, 11, 10, 10), Jin Huang's turn: (8, 8, 6, 5, 12, 12, 11, 10, 10)
Round 11: Lin Fan's turn: (8, 8, 6, 5, 12, 12, 11, 10, 9), Jin Huang's turn: (8, 8, 6, 5, 12, 12, 10, 10, 9)
Round 12: Lin Fan's turn: (8, 7, 6, 5, 12, 12, 10, 10, 9), Jin Huang's turn: (8, 7, 6, 5, 12, 11, 10, 10, 9)
Round 13: Lin Fan's turn: (8, 7, 6, 4, 12, 11, 10, 10, 9), Jin Huang's turn: (8, 7, 6, 4, 12, 10, 10, 10, 9)
Round 14: Lin Fan's turn: (8, 7, 6, 4, 12, 10, 10, 10, 8), Jin Huang's turn: (8, 7, 6, 4, 11, 10, 10, 10, 8)
Round 15: Lin Fan's turn: (8, 5, 6, 4, 11, 10, 10, 10, 8), Jin Huang's turn: (8, 5, 6, 4, 11, 10, 10, 10, 7)
After playing so many rounds, Lin Fan started to see through the ending of the game. Not for all possible endings, but to certain options.
He immediately made his choice. "From Pillar 7, remove 1 ring!"
Round 16.
Game state: (8, 5, 6, 4, 11, 10, 9, 10, 7).
Lin Fan had a smile on his face because he could feel that he was progressing somewhere.
He internally calculated a few possibilities. For example, if, in the next move, Jin Huang chooses to remove 2 rings from Pillar 5, it is forced mate in 17.
With best defenses, the game will be like this:
Round 17: Lin Fan's turn: (8, 5, 6, 4, 9, 7, 9, 10, 7), Jin Huang's turn: (8, 4, 6, 4, 9, 7, 9, 10, 7)
Round 18: Lin Fan's turn: (8, 4, 5, 4, 9, 7, 9, 10, 7), Jin Huang's turn: (8, 4, 4, 4, 9, 7, 9, 10, 7)
Round 19: Lin Fan's turn: (8, 3, 4, 4, 9, 7, 9, 10, 7), Jin Huang's turn: (8, 3, 4, 4, 9, 6, 9, 10, 7)
Round 20: Lin Fan's turn: (8, 3, 4, 4, 9, 3, 9, 10, 7), Jin Huang's turn: (8, 3, 4, 4, 9, 3, 9, 10, 6)
Round 21: Lin Fan's turn: (8, 3, 4, 4, 9, 3, 9, 7, 6), Jin Huang's turn: (8, 3, 4, 4, 9, 3, 9, 6, 6)
Round 22: Lin Fan's turn: (5, 3, 4, 4, 9, 3, 9, 6, 6), Jin Huang's turn: (5, 3, 4, 4, 9, 3, 8, 6, 6)
Round 23: Lin Fan's turn: (5, 3, 4, 4, 9, 3, 7, 6, 6), Jin Huang's turn: (5, 3, 4, 4, 9, 3, 6, 6, 6)
Round 24: Lin Fan's turn: (5, 3, 4, 4, 9, 3, 3, 6, 6), Jin Huang's turn: (4, 3, 4, 4, 9, 3, 3, 6, 6)
Round 25: Lin Fan's turn: (4, 3, 4, 4, 9, 3, 3, 3, 6), Jin Huang's turn: (4, 3, 4, 4, 9, 3, 3, 2, 6)
Round 26: Lin Fan's turn: (4, 2, 4, 4, 9, 3, 3, 2, 6), Jin Huang's turn: (2, 2, 4, 4, 9, 3, 3, 2, 6)
Round 27: Lin Fan's turn: (2, 2, 4, 4, 9, 3, 3, 2, 5), Jin Huang's turn: (2, 2, 3, 4, 9, 3, 3, 2, 5)
Round 28: Lin Fan's turn: (2, 2, 3, 3, 9, 3, 3, 2, 5), Jin Huang's turn: (2, 2, 3, 3, 8, 3, 3, 2, 5)
Round 29: Lin Fan's turn: (2, 2, 3, 3, 7, 3, 3, 2, 5), Jin Huang's turn: (2, 2, 3, 3, 6, 3, 3, 2, 5)
Round 30: Lin Fan's turn: (2, 2, 3, 3, 3, 3, 3, 2, 5), Jin Huang's turn: (2, 2, 3, 2, 3, 3, 3, 2, 5)
Round 31: Lin Fan's turn: (2, 2, 3, 1, 3, 3, 3, 2, 5), Jin Huang's turn: (2, 2, 3, 1, 3, 2, 3, 2, 5)
Round 32: Lin Fan's turn: (2, 2, 3, 1, 3, 1, 3, 2, 5), Jin Huang's turn: (2, 2, 3, 1, 3, 1, 3, 2, 4)
Round 33: Lin Fan's turn: (1, 2, 3, 1, 3, 1, 3, 2, 4)
Jin Huang has no legal move left, and she loses!
In this position, Jin Huang had only 1 legal move that saves her.
Pillar 2 (Remove 1 ring): Jin Huang is defeated by optimal counterplay in 24 rounds.
Pillar 2 (Remove 2 rings): Jin Huang is defeated by optimal counterplay in 24 rounds.
Pillar 5 (Remove 2 rings): Jin Huang triggers a lethal array trap, falling to optimal counterplay in 34 turns (as seen in the above case).
Pillar 7 (Remove 1 ring): Jin Huang is defeated by optimal counterplay in 24 rounds.
Pillar 7 (Remove 2 rings): Jin Huang is defeated by optimal counterplay in 24 rounds.
Pillar 9 (Remove 1 ring): Jin Huang is defeated by optimal counterplay in 25 rounds.
Pillar 9 (Remove 2 rings): Jin Huang is defeated by optimal counterplay in 23 rounds.
The only move that saves her is to remove 1 ring from Pillar 5.
Jin Huang glanced at Lin Fan and giggled. "Did you see something?"
She pointed her fingers to the 5th pillar and smiled. "Remove Ring 1!"