Can You Outsmart a Rogue AI? A Logic Puzzle

Imagine a world where a hostile artificial intelligence, known as NIM, controls all computers. You are humanity's last hope, tasked with shutting it down. You find yourself in NIM's secret lab, floating on a raft atop 25 stories of electrified water. Your mission: lower the water level to zero to activate a manual override and save the world.

But there's a catch. NIM is aware of your presence and can also manipulate the water level. You can lower the water by 1, 3, or 4 stories at a time using a remote, while NIM can do the same through a trapdoor. The control alternates between you and NIM, and neither of you can skip a turn. If NIM lowers the water to zero first, your mission fails.

Can you devise a strategy to ensure your victory against this cunning AI?

The Deadly Duel: Rules of Engagement

• Starting water level: 25 stories
• Your objective: Lower the water level to 0
• NIM's objective: Lower the water level to 0
• Possible moves: Lower the water level by 1, 3, or 4 stories
• Turns alternate, and no turn can be skipped.
• The water level cannot go below 0.
• NIM plays optimally to maximize its chances of success.
• You go first.

Cracking the Code: Working Backwards

The key to solving this puzzle lies in thinking strategically and working backward from the desired outcome. You want to be the one to bring the water level to 0. This means you need to leave NIM with a water level of 1, 3, or 4 on its turn. Why? Because from those levels, you can always win on your next turn.

Consider the water level at 2. If it's NIM's turn, their only option is to lower the water by 1, leaving you to win. This illustrates the concept of "losing" and "winning" levels.

• Losing Levels: No matter what move is made, the player starting on this level is destined to lose.
• Winning Levels: The player starting on this level can either win immediately or force their opponent onto a losing level.

1, 3, and 4 are winning levels because you can immediately win. 5 and 6 are also winning levels because you can force NIM to 2.

What about 7? Any move from 7 will leave NIM on a winning level, making 7 a losing level.

Identifying the Pattern

By continuing this analysis, a pattern emerges. If you start your turn 1, 3, or 4 levels above a losing level, you're on a winning level. Otherwise, you're destined to lose. Levels 8 through 11 behave identically to 1 through 4. This is because a level's status (winning or losing) is determined by the levels 1, 3, and 4 stories below it.

This pattern reveals a crucial insight: losing levels occur at multiples of 7 and two greater than multiples of 7.

The Winning Strategy

Starting at level 25, your goal is to ensure NIM always starts its turn on a losing level. If NIM ever starts on a winning level, it's game over for you.

Therefore, your only viable move on the first turn is to lower the water level by 4 stories, bringing it to 21. From there, no matter what NIM does, you can always force it onto a losing level until you reach 0 and trigger the manual override.

By consistently forcing NIM onto losing levels, you secure your victory and save the world from the clutches of the rogue AI.