How to calculate distances on a hexagonal grid

We label the cells of the grids thus:-

We can refer to a cell as [x,y]. We assume the distance between the centres of two adjacent cells is 1

1. 'Cell by cell' moves

What is the minimum distance that a (chess) king would travel if it started at cell [x0,y0] and ended at [x1,y1]? We assume the king moves one cell at a time in any direction, and that x1>=x0 (swap the cells if x1<x0)

Answer:-
   IF y1>y0 THEN Distance=x1-x0+y1-y0
   ELSEIF x0+y0>x1+y1 THEN Distance=y0-y1
   ELSE Distance=x1-x0

The 3 cases correspond to the king moving
   NorthEast then Up
   SouthEast then Down
   NorthEast then SouthEast or vice versa

2. 'As the crow flies'

The distance between the centres of cells [x0,y0] and [x1,y1] is given by:-

Should you wish to calculate the distance (as the crow flies) without using a coordinate system - simply moving y cells in the y-direction then x cells in the x-direction (or vice versa), the formula simplifies to:-

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