Hang them all so it hangs level.
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A shape weighs its number of sides. The triangle weighs three, the square four, up to the octagon at eight. The five-pointed star weighs ten and the six-pointed star twelve — its outline has twelve strokes, not six. Each weight is printed on the shape; 123 hides them if you would rather count for yourself.
Notches multiply. A beam hangs from one of its notches, and the distance out multiplies the weight — a square hung three notches from the middle hangs like a twelve, not a four. Two shapes on the same string are simply stacked — which is on top makes no difference.
A beam hanging from a beam is one weight. Add up every shape below it and treat that total as a single weight hanging there. So any beam is only ever two numbers being compared. Each beam still has to come level in its own right, which is what lets you work down from the top.
Hang them all so every beam is level. Tap a shape, then tap a hook. Tap a hanging shape to take it back down.
The way in
The whole mobile weighs whatever the tray weighs — and that is on screen before you hang a thing. A beam that splits its load two-to-one has to carry twice as much on the short arm, so both its arms are decided the moment you know the total. Work down from the top and most of the mobile falls out on its own.
Every beam hangs the way the shapes on it actually pull it, so an arm with nothing on it yet rides up. A beam turns red once what's left in the tray can no longer level it, and green when it is finished and level.
A hint hangs one shape for you — the next one you could have worked out. It can't be taken back down.