Turn every knob flat so the shuttle can slide out of the case. The same mathematics as the Chinese rings and Spin-Out.
Knobs
Start
How it works. A knob can only turn under the notch. To get knob k there, the shuttle slides right: knob k − 1 lands in the pocket, and knobs 1 to k − 2 slide out through the end of the case, which only fits flat knobs. Each knob has three curved indents and one bump. Upright, an indent faces the knob on each side, so it never gets in their way. Flat, its bump faces the knob to its left and blocks it from swinging, while an indent still faces the knob on its right. So knob k can swing only if knob k − 1 is upright. Knob 1 has no neighbor on its right and can always turn.
Where you are on the only path through every position
Try the puzzle first. When you want to see how many moves you have left and why, check “Show the secret” above the puzzle.
Write 1 for an upright knob and 0 for a flat one. Every legal move changes exactly one digit, and all 2ⁿ possible rows of digits lie along a single path, so there is never a choice about which way is forward. To find how many moves you still need, start at the far left and count upright knobs as you go: write 1 each time your count so far is odd and 0 each time it is even. Read the result as a binary number.
The Chinese rings, a loop of rings threaded on a wire, is centuries old; the Italian mathematician Gerolamo Cardano described it around 1550. In 1872 the Frenchman Louis Gros worked out that its positions follow a binary pattern in which each step changes one digit. Engineer Frank Gray rediscovered the pattern at Bell Labs and patented it in 1953, which is why it is called the Gray code. Computers and rotary encoders still use it today, because a reading that changes only one bit at a time can never be caught halfway between two values.
William Keister turned the rings into a sliding plastic puzzle, patented in 1972 and sold as Spin-Out, now made by ThinkFun. Holding one in your hands is even better than this version.