Ball Lock Puzzle

myminifactory

Here's an entertaining little challenge that I'm confident you've never encountered before because I just created it. I've been calling it the z=xy puzzle; read on to discover why. It is a more complex version of my 2 piece cube puzzle, but now including a magic ball lock. It is part puzzle, part magic trick, part math lesson. The theory that makes it work comes from pure mathematics. Once the cube is closed and locked by rotating the ball, the cube is held closed by invisible mathematical forces. Normally, a sphere would not be retained in a hemispherical cavity, but not in this case. What's behind this phenomenon? You start with the cube in two halves, each side holding half of a ball. Show your audience member how to view the inside of the cube and point out that the inner surfaces are all smooth with no overhangs to hold the cube together. Slide the cube halves all the way together to form a solid cube and then apart again to demonstrate that the cube opens and closes freely. Now close the cube and hand it to an audience member with instructions to pull it apart. They can't. The Secrets There are two secrets, 1) secretly rotate the ball from the back side of the cube as you hand it to the audience member: this locks the cube shut, 2) invisibly flick the cube itself 180 degrees from its original position when opened. Combine these two actions into one motion as you hand over the cube. The unsuspecting audience member will try to pull the cube open just like you did but won't be able to because they'll be pulling the pieces closed instead of open after the rotation. If they figure that out, the cube will still not open because the ball lock holds it shut unless they rotate the ball in the correct orientation on two axes and pull on the right cube faces. How it Works There are two facts that make it work. 1) The ball is bisected symmetrically but in a special way (z=xy), allowing only one axis for separation. 2) The profile of the hemisphere viewed from its pole is non-circular, leading to a paradox in the audience's mind. Each hemisphere locks into the socket on the opposite side of the cube unless rotated correctly. The combination of these two facts means that the cube will only open when the ball separation axis aligns with the cube separation axis and the ball is rotated in the correct direction for release. The z=xy Puzzle Some of you may have noticed that the cool saddle-like surface bisecting the sphere is a hyperbolic paraboloid, z=xy. Here's a practical application of something you learned about in school. Who knew? Applied mathematics meets 3D printing. Printing I've provided the complete puzzle in one file for single-color printing and separate files for each piece to print in different colors at your request. UPDATE: I added another version with a hollow core, so you can hide something inside. Print at 0.2 mm layer thickness with support (required for the ball only) and ensure plenty of fan airflow to smooth out overhangs on the bottom of the ball. You may need to use sandpaper to round out the bottom if it doesn't come out perfectly spherical. The outside of the ball needs to be smooth and perfectly spherical so it turns freely inside its housing. Assembly Insert the hemispheres into the housing with their saddle high side pointing towards the open sides, then rotate them 90 degrees so that the saddle low points face the open sides. This is the default UNLOCKED position, allowing the cube to go together or come apart easily. Slide the two puzzle halves together while rotating one of them 90 degrees out of phase. Any rotation of the ball will lock the puzzle, so be sure to observe it closely before moving it. The layer lines on the ball can help guide you in finding its correct position. Once moved, put it back in its original position or rotated by 180 degrees to open the puzzle. Get it from Shapeways If you prefer not to print one yourself or would like a metal version, I have them available on Shapeways at my cost. Note that the Shapeways model has been modified to meet their minimum wall thickness requirements, so it looks slightly different but uses the same idea. You'll need to order two puzzles if getting it from Shapeways since it's only half the puzzle. On the plus side, you can get each side in its own color.