One simple algorithmic solution to the puzzle is as follows.

1. Define \(\Delta x = |x_A - x_B|, \Delta y = |y_A - y_B|, \Delta z = |z_A - z_B|\).

2. Sort these quantities: \[[\text{min, med, max}] = \text{sort}[\Delta x, \Delta y, \Delta z].\]

3. Set \(n_{111} = \text{min}\), \(n_{110} = \text{med-min}\), \(n_{100} = \text{max-med}\).

4. The minimum distance is given by \[d_{min} = \sqrt{3} n_{111} + \sqrt{2} n_{110} + n_{100}\]

As an example consider A = (5, 3, 2) and B = (4, -1, 2).

1. \(\Delta x = 1, \Delta y = 4, \Delta z = 0\).

2. [min, med, max] = [0, 1, 4]

3. \(n_{111} = 0\), \(n_{110} = 1\), \(n_{100} = 3\).

4. Minimum distance is \(\sqrt{2} + 3\).

1. Define \(\Delta x = |x_A - x_B|, \Delta y = |y_A - y_B|, \Delta z = |z_A - z_B|\).

2. Sort these quantities: \[[\text{min, med, max}] = \text{sort}[\Delta x, \Delta y, \Delta z].\]

3. Set \(n_{111} = \text{min}\), \(n_{110} = \text{med-min}\), \(n_{100} = \text{max-med}\).

4. The minimum distance is given by \[d_{min} = \sqrt{3} n_{111} + \sqrt{2} n_{110} + n_{100}\]

As an example consider A = (5, 3, 2) and B = (4, -1, 2).

1. \(\Delta x = 1, \Delta y = 4, \Delta z = 0\).

2. [min, med, max] = [0, 1, 4]

3. \(n_{111} = 0\), \(n_{110} = 1\), \(n_{100} = 3\).

4. Minimum distance is \(\sqrt{2} + 3\).

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