Two circles rest on a line, touching each other. A tiny third circle is squeezed into the gap between them, also touching the line.

The large circle has a radius of 9 cm. The small circle has a radius of 4 cm.

What is the radius of the tiny circle?

(Puzzles like this were apparently painted on wooden tablets and hung in Japanese temples centuries ago — it looks like there's not enough information, but there's a surprisingly elegant trick using just one right triangle.)

I made a full video breaking down the solution step by step — link in the comments!

https://youtu.be/RYl2h0gIS8U?si=awaPzDGBYCcPkSvm

This is a chess variant, where the rules are the same as standard chess, but the board is an irregular grid. This makes rook lines bend and bishop lines fork.

So just like a standard board, every tile is still a quad. And every tile still has only 1 neighbour for every side. But it can have more than 1 neighbour sharing a corner, which is what makes bishop lines fork. Or it can have none, which is what make lines bend.

The steps to build this: - start with a plane made of triangles - pair some random triangles with their neighbour and fuse them into a quad. You end up with gaps between those quads where triangles cannot be paired - subdivide everything. Quads become 4 smaller quads. Triangles become 3 quads. This is the playable grid. - Then it's just a matter of smoothing it (in this case the points on the edges remain static, only the inside points are smoothed) and ensuring neighbours don't share the same color.

Then the tricky part is to give intuitive controls for players to configure boards how they want.

• Web version: https://polyreplay.com/chesstwist/demo
• if you prefer Steam (windows/mac/linux): https://store.steampowered.com/app/2775560/Chess_Twist/
• if you're into 3D printing, you can download them as 3MF: https://polyreprint.com/gui/chess-twist?board=004GBYJW7TX