Imagine a scenario where your research team has unearthed a prehistoric virus, preserved in the permafrost, and isolated it for study. After a long night at the lab, just as you’re about to leave, an earthquake strikes, cutting off the power. The emergency generators kick in, and an alarm confirms your worst fears: all the sample vials have shattered. The virus is contained for now, but unless you act quickly, the vents will soon open, releasing a deadly airborne plague.
Equipped with a HazMat suit, you prepare to tackle the challenge. The lab is structured as a four by four compound of 16 rooms, with an entrance in the northwest corner and an exit in the southeast. Each room is connected to its neighbors by an airlock, and the virus has spread to every room except the entrance. To neutralize the threat, you must enter each contaminated room and activate its emergency self-destruct switch. However, there’s a catch: once you enter a contaminated room, you can’t leave without activating the switch, and once activated, you can’t re-enter that room.
As you sketch possible routes on paper, you realize that reaching the exit without missing a room seems impossible. This predicament is akin to the Hamiltonian path problem, named after the 19th-century Irish mathematician William Rowan Hamilton. The challenge is to find a path that visits every point exactly once. This problem, classified as NP-complete, is notoriously difficult for large graphs. Although any solution can be easily verified, finding one or determining its existence lacks a reliable formula or shortcut. Even computers struggle to solve such problems reliably.
This particular puzzle adds a twist to the Hamiltonian path problem: you must start and end at specific points. However, a true Hamiltonian path isn’t feasible with these endpoints. The rooms form a grid with an even number of rooms on each side, making it impossible to start and end a Hamiltonian path in opposite corners. Consider a checkerboard grid with an even number of squares on each side. Every path alternates between black and white squares, and since the grid has an even total number of squares, a path starting on black must end on white, and vice versa. Yet, opposite corners in such grids are the same color, making it impossible to start and end on opposite corners.
All hope seems lost until you notice an important exception in the rules. Once you activate the switch in a contaminated room, it’s destroyed, and you can’t return. However, the entrance wasn’t contaminated, allowing you to leave it once without pulling the switch and return after destroying two other rooms. This return trip provides four options for a successful route, with similar options if you destroy another room first. Congratulations, you’ve averted an epidemic of apocalyptic proportions!
After such a stressful episode, perhaps it’s time for a change. Maybe that recent job offer to become a traveling salesman is worth considering. After all, you’ve just navigated a complex puzzle and saved the world from a prehistoric virus.
Using graph paper or a digital drawing tool, recreate the 4×4 lab layout described in the article. Mark the entrance and exit, and simulate the process of navigating through the contaminated rooms. Try to find a path that visits each room exactly once, adhering to the rules mentioned. Discuss your findings with your classmates.
Research the Hamiltonian path problem further. Create a presentation that explains the problem, its significance in computer science, and why it is classified as NP-complete. Include examples of different types of graphs and whether they have Hamiltonian paths. Present your findings to the class.
Draw a checkerboard grid with an even number of squares on each side. Using colored markers or digital tools, attempt to trace a path that starts and ends on opposite corners, visiting each square exactly once. Reflect on why this is impossible and relate it to the concepts discussed in the article.
In groups, role-play the scenario described in the article. Assign roles such as the lead scientist, lab technician, and emergency response coordinator. Discuss and act out the steps you would take to neutralize the virus threat. Reflect on the importance of teamwork and quick decision-making in emergency situations.
Write a short story or a diary entry from the perspective of a scientist who has just averted the virus outbreak. Describe the emotions, challenges, and thought processes involved in solving the puzzle. Share your story with the class and discuss the different perspectives and creative approaches taken by your peers.
virus – A microscopic infectious agent that can only replicate inside the living cells of an organism. – The study of how a virus spreads can help scientists develop vaccines to prevent outbreaks.
lab – A controlled environment where scientific experiments and research are conducted. – In the biology lab, students observed the effects of different variables on plant growth.
path – A sequence of steps or a route taken to solve a mathematical problem or to understand a biological process. – The path to finding the area of a triangle involves using the formula A = 1/2 * base * height.
problem – A question or situation that requires a solution, often involving mathematical calculations or biological concepts. – The math problem required students to calculate the volume of a cylinder using its radius and height.
rooms – Enclosed spaces within a building, often used for specific purposes such as classrooms or laboratories. – The school has several rooms dedicated to science experiments, including a chemistry lab and a biology room.
grid – A network of lines that cross each other to form a series of squares or rectangles, often used in graphing mathematical functions. – The students plotted the data points on a grid to visualize the relationship between temperature and enzyme activity.
corners – The points where two lines meet, often used in geometry to define the vertices of shapes. – In geometry class, we learned how to calculate the angles formed at the corners of a triangle.
squares – A geometric shape with four equal sides and four right angles, often used in mathematical problems. – The area of a square can be calculated by squaring the length of one of its sides.
solution – The answer to a mathematical problem or the process of solving a problem in biology. – After several attempts, she finally found the solution to the complex algebraic equation.
epidemic – A widespread occurrence of an infectious disease in a community at a particular time. – The recent epidemic highlighted the importance of vaccination and public health measures to control disease spread.
