Cipher & Cinder

An instance shows a hanging mobile: a fixed complete binary tree of 7 rods over 8 hooks, where each rod carries an integer arm pair and two of the rods have had their arm pair erased, together with a sorted bag of 8 weights, one per hook. The answer fills in which weight hangs on each hook and the two erased arm pairs, and is correct when the whole bag is used exactly once, every rod balances (left total x left arm = right total x right arm), and every arm pair is in lowest terms.

A 7 by 7 grid holds walls and six coloured territories, drawn by growing each colour outwards one step at a time from a single hidden starting cell, with every cell going to whichever colour reached it first and ties going to the lower colour number. Given only the finished map, name the starting cell of each of the six colours.

A 3-by-4 lattice of junctions is joined by 17 one-way pipes, drawn on a 5-by-7 board where junction cells and pipe cells alternate. Each junction cell shows how much liquid it adds or removes, each pipe cell shows how much flows through it, twelve pipe cells have been erased to '?', and the task is to write back a flow of 0 to 4 in every erased cell so that every junction balances exactly.