TED-Ed Video Lesson Transcript:
You’re on the trail of a werewolf that’s been terrorizing your town. After months of detective work, you’ve narrowed your suspects to one of five people: the mayor, the tailor, the baker, the grocer, or the carpenter.
You’ve invited them to dinner with a simple plan: you’ll slip a square of a rare werewolf antidote into each of their dinners.
Unfortunately, your pet goat just ate four of the squares, and you only have one left. Luckily, the remaining square is 50 grams, and the minimum effective dose is 10 grams.
If you can precisely divide the square into fifths you’ll have just enough antidote for everyone You’ll have to use a laser-cutting tool to cut up the square; every other means available to you isn’t precise enough.
There are 8 points that can act as starting or ending points for each cut. To use the device, you’ll have to input pairs of points that tell the laser where to begin and end each cut, and then the laser executes all the cuts simultaneously.
It’s okay to cut the square into as many pieces as you want, as long as you can group them into 10 gram portions.
But you can’t fold the square or alter it otherwise, and you only get one shot at using the laser cutter.
The full moon is rising, and in a moment someone will transform and tear you all apart unless you can cure them first.
How can you divide the antidote into perfect fifths, cure the secret werewolf, and save everyone?
Pause the video now if you want to figure it out for yourself. Answer in 3. Answer in 2. Answer in 1.
When it comes to puzzles that involve cutting and rearranging, it’s often helpful to actually take a piece of paper and try cutting it up to see what you can get.
If we cut BF and DH we’d get fourths, but we need fifths. Maybe there’s a way to shave a bit off of a quarter to get exactly one fifth.
Cutting BE looks good at first, but that last cut takes a off a quarter of a quarter, leaving us with a portion of 3/16: just smaller than a fifth, and not enough to cure a werewolf.
What if we started with BE instead? That would also give us a quarter.
And is there a way to shave just a bit more off? Both DG and CH look promising. If we make one more cut, from A to F, we may start to notice something. With these four cuts—from B to E, D to G, F to A, and H to C—we’ve got four triangles and a square in the middle.
But the pieces that make each triangle can also be rearranged to make a square identical to the middle one.
This means that we’ve split the antidote into perfect fifths! What’s interesting about this sort of problem is that while it’s possible to solve it by starting from the geometry, it’s actually easier to start experimenting and see where that gets you.
That wouldn’t be as viable if the square had, say, 24 cut points, but with just 8 there are only so many reasonable options. You secretly dose each of the townspeople as the full moon emerges in the sky.
And just as you do, a terrible transformation begins. Then, just as suddenly, it reverses.
Your measurements were perfect, and the people and animals of the town can rest a little easier.