Ball Lock Puzzle

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Ball Lock Puzzle I've created a novel little puzzle that's sure to intrigue you - one that I call the z=xy puzzle; read on to discover why. It's a more intricate version of my 2-piece cube puzzle, now including a magic ball lock. This puzzle combines elements of math, magic tricks, and educational lessons in mathematics. The mathematical theory behind it comes from pure math. Once the cube is closed and locked by rotating the ball, the cube remains shut due to invisible forces that defy conventional logic. Normally, a sphere wouldn't be retained within a hemispherical cavity, but not in this case. What makes it possible? You start with the cube in two halves, each side holding half of a ball. Show your audience the inside of the cube and point out how its 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 it opens and closes freely. Now close the cube and hand it to an audience member with instructions to pull it apart; they won't be able to. The Secrets There are two secrets: 1) secretly rotate the ball from the back side of the cube as you hand it over, locking the cube shut; 2) invisibly flick the cube itself 180 degrees from its original position when you opened it. Combine these motions into one action as you hand it over. The unsuspecting audience member will try to pull the cube open like you did but won't be able to because they'll be pulling the pieces closed, not open, after the rotation. If they figure that out, the cube still won't 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 this puzzle work: 1) the ball is bisected symmetrically but in a special way (z=xy) that only allows its halves to be separated exactly along one axis; 2) the profile of the hemisphere viewed from the pole is non-circular, counterintuitively so, leading to a paradox in the audience's mind. Each hemisphere locks into the socket on the opposite side of the cube unless it's rotated in the correct direction. The combination of these two facts means that the cube will only open when its ball separation axis aligns with the cube separation axis and the ball is rotated in the one direction where it releases from the socket on the opposite side. 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. This puzzle offers a practical application of a mathematical concept learned in school - who knew? Applied mathematics meets 3D printing. Printing I've provided the complete puzzle in one file for those printing in one color to use that file. For those wanting separate pieces, I've included files so each piece can be printed in a different color. UPDATE: at raver1975's request, I've added another version of the ball with a hollow core, allowing something to be hidden inside. Print at 0.2 mm layer thickness, using support (required for the ball only). Make sure to use plenty of fan so overhangs on the bottom of the ball come out smooth; you may need to take sandpaper and round out the bottom if it doesn't come out perfectly round. The outside of the ball needs to be smooth and perfectly spherical so it turns freely inside the housing. Assembly The hemispheres must be inserted into the housing properly for the puzzle to work. Insert the hemispheres with them rotated 90 degrees out of phase, in their default UNLOCKED position - this is the only position where the cube will go together or come apart. Any rotation of the ball will lock the puzzle, so take a close look before moving it; you can tell where it goes by looking closely at layer lines on the ball. Once you move the ball, put it back in its exact same position (or rotated by 180 degrees) to open the puzzle. Get it on Shapeways If you'd like one of these puzzles but don't have a 3D printer yet or prefer one made in metal, I offer them available on Shapeways at my cost . Note that the Shapeways model has been modified to meet their minimum wall thickness requirement, so looks a little different; it uses the same idea. You'll need to order two if you get it from Shapeways, since it's only half the puzzle. On the plus side, you can get each side in its own color. The home print version actually has tighter seams but in plastic, you can force it open.