A shooting star crashes onto Earth and a hideous blob emerges. It creeps and leaps, it glides and slides. It’s also unstoppable: no matter what you throw at it, it just re-grows and continues its rampage. The only way to save the planet is to cut the entire blob into precise acute triangles while it sleeps, rendering it inert. Can you stop the blob from destroying the planet? Dan Finkel shows how.
A shooting star crashes on Earth, and a hideous blob emerges. It creeps and leaps, it glides and slides. It’s also unstoppable: weapons, fire, extreme temperatures… no matter what you throw at it, it just regrows and continues its rampage.
Its expansion is breathtaking; it doubles in size every hour. But there’s one opportunity: after each hour, it goes to sleep, forming itself into a flat triangle and resting for a few minutes before it begins eating and growing again.
Your only chance to save the planet involves a satellite-mounted nano-fission ray that can cut through the blob. When the blog is active it heals itself within seconds. However, when you break the sleeping blob into two triangles, you make a critical discovery. The acute triangle portion, with all angles less than 90 degrees, is inert. It never “wakes up.” The obtuse triangle, which has an angle greater than 90 degrees, wakes up as usual and keeps growing. Similar experiments show that all shapes other than acute triangles, including right triangles, will also wake up.
For the next few minutes, the blob is sleeping in its obtuse triangle form. You can make clean, straight-line cuts between any two points on or inside the triangle. But you’ll only have time to make 7 cuts while the satellite is above you. By the time it completes its orbit and returns, the blob will have consumed the entire world, if even a single portion that will wake up remains.
How can you cut the blob entirely into acute triangles and stop it from destroying the planet?
While this seems doable at first, there’s a hidden difficulty when it comes to avoiding obtuse and right angles. Every time you make a cut that reaches an edge, it either makes an acute and an obtuse angle, or two right angles.
That makes it seems like you’re doomed to keep creating obtuse angles. But as with so many of life’s problems, we can look to pizza for inspiration. Imagine squaring off the outside of a pizza, so that instead of a circle, it’s an octagon. When we cut it into slices, each of the eight triangles is acute. This works with larger polygons too. Importantly, it also works for some polygons with fewer sides, including heptagons, hexagons, and pentagons. That’s good news,
because if you cut off the sharp corners of the blob triangle, a pentagon is exactly what you’ll be left with. And just like a pizza, you can cut the blob pentagon into five acute triangles. That’s 7 cuts, and it renders the blob completely inert. You’ve saved the day! Now you just need to figure out what to do with all of these giant, practically indestructible triangles.