Imagine a super-smart artificial intelligence named NIM has taken control of the world’s computers. You’re the only one who can stop it, and you have just one chance to succeed. You’ve managed to sneak into NIM’s secret lab, and now you’re on a raft floating on top of 25 stories of electrified water. Your mission is to lower the water level to zero using a remote control, which will allow you to shut down NIM and save the day.
But there’s a catch! NIM knows you’re there and can also lower the water level. If NIM gets the water to zero first, you’ll be sucked out of the lab, and the mission will fail. You and NIM take turns lowering the water, and neither of you can skip a turn. Each turn, you can lower the water by exactly 1, 3, or 4 stories. The goal is to be the one who lowers the water to exactly zero.
To win, you need to think ahead and plan your moves carefully. The key is to work backward from where you want to end up. You want to be the one to lower the water to zero, which means you need the water level to be at 1, 3, or 4 when it’s your turn. If the water level is at 2, your only option is to lower it by 1 story, which would allow NIM to win on its next turn.
Some water levels are “losing” levels, like 2, where no matter what you do, you’ll lose. Other levels are “winning” levels, where you can either win or force NIM into a losing position. Levels 1, 3, and 4 are winning levels, and so are 5 and 6, because from there, you can send NIM to level 2.
What about level 7? From level 7, any move you make will send NIM to a winning level, making 7 a losing level. By analyzing the water levels this way, you can see a pattern: if you start your turn 1, 3, or 4 levels above a losing level, you’re on a winning level. Otherwise, you’re likely to lose.
You can apply this pattern all the way up to level 25. Notice that levels 8 through 11 behave like levels 1 through 4. This means level 12 is like level 5, level 13 is like level 6, and level 14 is like level 7. Losing levels are multiples of 7 and two more than multiples of 7.
Starting from level 25, you must ensure NIM always starts its turn on a losing level. If NIM starts on a winning level even once, you’ll lose. So, on your first turn, lower the water by 4 stories. No matter what NIM does, you can keep putting it on losing levels until you reach zero and activate the manual override. With this strategy, you can outsmart NIM and save the world!
Now that you’ve averted the crisis, you can relax and enjoy some less stressful activities. Good luck, and happy problem-solving!
Imagine you’re in the lab facing NIM. Create a simple board game with a path of 25 spaces representing the water levels. Use tokens to represent you and NIM. Take turns with a classmate to lower the water by 1, 3, or 4 spaces. Your goal is to be the one to reach zero first. This will help you understand the concept of winning and losing levels.
Form small groups and discuss different strategies to ensure you always leave NIM on a losing level. Share your ideas and see if you can come up with a foolproof plan. This activity will help you think critically and collaboratively about problem-solving.
Write a short story about your adventure in NIM’s lab. Include the strategies you used to outsmart the AI. Share your story with the class and see how others approached the challenge. This will enhance your understanding of the concept while allowing you to be creative.
Create math puzzles based on the winning and losing levels. For example, if you start at level 18, what moves should you make to ensure victory? Challenge your classmates to solve your puzzles. This will reinforce your understanding of the mathematical patterns in the game.
Role-play as either yourself or NIM in a debate. Argue why your strategy is the best to win the game. This activity will help you articulate your understanding of the game mechanics and develop persuasive communication skills.
A hostile artificial intelligence called NIM has taken over the world’s computers. You’re the only person skilled enough to shut it down, and you’ll only have one chance. You’ve broken into NIM’s secret lab, and now you’re floating in a raft on top of 25 stories of electrified water. You’ve rigged up a remote that can lower the water level by ejecting it from grates in the sides of the room. If you can lower the water level to 0, you can hit the manual override, shut NIM off, and save the day. However, the AI knows that you’re here, and it can lower the water level too, by sucking it through a trapdoor at the bottom of the lab. If NIM is the one to lower the water level to 0, you’ll be sucked out of the lab, resulting in a failed mission.
Control over water drainage alternates between you and NIM, and neither can skip a turn. Each of you can lower the water level by exactly 1, 3, or 4 stories at a time. Whoever gets the level exactly to 0 on their turn will win this challenge. Note that neither of you can lower the water below 0; if the water level is at 2, then the only move is to lower it 1 story. You know that NIM has already computed all possible outcomes of the contest and will play in a way that maximizes its chance of success. You go first. How can you survive and shut off the artificial intelligence?
You can’t leave anything up to chance – NIM will take any advantage it can get. And you’ll need to have a response to any possible move it makes. The trick here is to start from where you want to end and work backwards. You want to be the one to lower the water level to 0, which means you need the water level to be at 1, 3, or 4 when control switches to you. If the water level were at 2, your only option would be to lower it 1 story, which would lead to NIM making the winning move.
If we analyze the water levels, we can see a simple principle at play: there are “losing” levels like 2, where no matter what the player does, they’ll lose. And there are winning levels, where whoever starts their turn there can either win or leave their opponent with a losing level. So not only are 1, 3, and 4 winning levels, but so are 5 and 6, since you can send your opponent to 2 from there.
What about 7? From 7, all possible moves would send your opponent to a winning level, making this another losing level. We can continue this analysis up the lab in this way. If you start your turn 1, 3, or 4 levels above a losing level, then you’re at a winning level. Otherwise, you’re destined to lose.
You could continue this all the way to level 25. But as a shortcut, you might notice that levels 8 through 11 are colored identically to 1 through 4. Since a level’s status is determined by the levels 1, 3, and 4 stories below it, this means that level 12 will be the same as level 5, 13 will match 6, 14 will match 7, and so on. In particular, the losing levels will always be multiples of 7 and two greater than multiples of 7.
Now, from your original starting level of 25, you have to make sure your opponent starts on a losing level every single turn—if NIM starts on a winning level even once, it’s game over for you. So your only choice on turn 1 is to lower the water level by 4 stories. No matter what the AI does, you can continue giving it losing levels until you reach 0 and trigger the manual override. And with that, the crisis is averted. Now, back to a less stressful kind of surfing.
Water – A liquid substance that can be measured in volume and used in mathematical problems involving capacity or flow rates. – In our math class, we calculated how much water would fill a swimming pool by using the formula for volume.
Level – A specific height or position, often used in geometry or measurement to describe a flat surface or equal height. – We used a ruler to ensure that the table was level before starting our geometry project.
Losing – The act of not achieving a desired outcome, often used in probability to describe an unfavorable result. – In our probability game, the chance of losing was higher if we picked a red card from the deck.
Winning – The act of achieving a desired outcome, often used in probability to describe a favorable result. – By calculating the odds, we increased our chances of winning the math competition.
Strategy – A plan of action designed to achieve a specific goal, often used in problem-solving or logical reasoning. – Our strategy for solving the complex equation involved breaking it down into simpler parts.
Moves – Actions taken to change position or state, often used in geometry or logic puzzles. – In the chess puzzle, we had to think carefully about our moves to checkmate the opponent.
Plan – A detailed proposal for achieving something, often used in solving mathematical problems or projects. – We created a plan to divide the tasks equally among team members for our math project.
Analyze – To examine in detail in order to understand or interpret, often used in solving complex problems or data sets. – We had to analyze the graph to determine the trend in the data over time.
Pattern – A repeated or regular arrangement, often used in sequences or geometric designs. – We identified the pattern in the sequence of numbers to predict the next term.
Control – The power to influence or direct variables, often used in experiments or mathematical models. – In our math experiment, we kept the temperature constant to control the results accurately.
