Background
Read James Propp’s “Solving the Flat Cube” first. This page uses the convention that one move is exactly one 60° turn.
Play with it
Claim
God’s number is the largest optimal solution length over all reachable positions. Propp says the colored Flat Cube’s God’s number is unknown and gives a lower bound of 27. With one 60° turn counted as one move, the potential argument here gives a lower bound of 81.
Define a potential for each color (see below). The solved position has total potential −54, the 81-move position has total potential 189, and each move changes the total potential by at most 3 in absolute value.
So at least 81 moves are necessary. And 81 moves are achievable by adding each of the 27 cubes with a 180° rotation, counted as three consecutive 60° moves. Therefore this configuration has exact distance 81.
Potentials
For a cube position (x,y,z), call its six faces X−, X+, Y−, Y+, Z−, Z+.
Potential formulas
X+: 3x − y − z + 3
Y−, Z−: 3x − y − z + 2
Y+, Z+: 3x − y − z + 1
Y+: 3y − z − x + 3
Z−, X−: 3y − z − x + 2
Z+, X+: 3y − z − x + 1
Z+: 3z − x − y + 3
X−, Y−: 3z − x − y + 2
X+, Y+: 3z − x − y + 1
An interior face can be described from either neighboring cube, and the formulas give the same value whichever side is used. Hover a numbered lozenge below to see the actual coordinates substituted both ways.
A cube-changing 60° turn changes the local patch {X−, Y−, Z−} ↔ {X+, Y+, Z+}. Each lozenge also changes orientation, so a minus-face can only transition to one of the two differently lettered plus-faces, and vice versa.
Every allowed double-arrow pair above changes that color’s potential by exactly 1 in absolute value. A move contains three lozenges, so |ΔΦ| ≤ 3.
Φ is the total potential across all faces.