The Secret of the Square Corner

A math mystery you solve yourself: discover it, prove it, use it, teach it 🕵️

PART 1

A strange discovery on the floor 👀

PART 2

The Square Machine 🔢

Was the floor a fluke, or does it work for every triangle with a square corner? Build your own. Drag the two round handles to change the triangle. First guess how many little tiles fit in the mystery gold square… then click the little tiles in the coral and teal squares. Each one flies over and lands in the gold square, so you can watch whether they fit.

PART 3

The Sliding-Puzzle Proof 🧩

Pythagoras couldn't check every triangle one by one, so he needed a single idea that covers all of them at once. Here it is. A square room with 4 identical triangle tables inside. Nothing gets added and nothing gets taken away, so however the tables slide around, the amount of empty gold floor stays the same. Its shape doesn't.
Your mission: drag the triangles until the empty floor becomes two straight squares. There's more than one way to do it, and any arrangement that works counts.

PART 4

The Water Proof 💧

Still not convinced? Fair enough. The two small squares are tanks full of water. Flip the whole thing upside-down and let gravity pour every drop into the big square. Make your prediction first: will it overflow, be half-empty… or fill it exactly?
🖐 You can also grab the picture and spin it yourself. The water always flows downhill.

PART 5

The superpower: measure with your brain 🧮

You proved the secret is true. Here's what it lets you do: work out the length of a slanty side without measuring it.

PART 6

Mission time 🦸

Now use the recipe on real problems, where a ruler can't reach. Three missions.
Your recipe: ① square both short sides ② add them ③ find the number that times itself makes the total. (5×5=25, 10×10=100, 13×13=169…)

PART 7

Now YOU are the teacher 🎓

You know four ways to show this to someone else. Teaching it is how you find out whether you really have it.

1. The Tile Story 🏛️

Draw a floor of square tiles. Shade half a tile, then draw the three squares like Pythagoras did.
Say: "Count the little triangles. Two and two in the small squares, four in the big one. Equal."

2. Counting Squares 🔢

Draw a triangle with sides 3 and 4 on grid paper and grow squares on each side.
Say: "9 tiles + 16 tiles = 25 tiles. The squares on the short sides exactly fill the big one!"

3. The Sliding Puzzle 🧩

Cut 4 identical paper right triangles. Put them in a square two different ways.
Say: "Same room, same tables, so the empty floor has to be equal. One big square, or the two small ones."

4. The Water Pour 💧

Describe (or rebuild!) the water experiment.
Say: "Pour the two small squares into the big one and it fills exactly. No spill, no gap."

Taught someone? Put your name on this: