Find the max distance at least one of 6 cars can cover if each car can cover 1000km and cars can recharge each other's batteries

Question

These are electric cars.

Edit: The efficient approach would take symmetry into account as shown by @R. J. Mathar

Answer 1

There is at least one simple symmetric solution (not necessarily the maximum, but optimization problems often have extrema at the symmetric choices): Let all 6 cars drive from the initial point (all range 1000) by 1000/6 so their remaining range is down to $5\times 1000/6$. Recharge 5 of the cars equally with the power of the 6th (the 6th is halted there) so we have 5 cars at distance 1000/6 from the origin with range 1000 left. After another distance 1000/5 recharge 4 of the cars with the power of the 5th and this leaves 4 cars at distance 1000/6+1000/5 with range 1000 left. Recursively ... the last car can reach $1000/6+1000/5+1000/4+1000/3+1000/2+1000 = 1000\times 49/20 = 2450$.