3 cell or 4 cell

Lets say you have to choose between two flashlights to take with you on a trip. Both have identical LS LED's and optics. One uses 3 aa batteries and has a series resistor selected to provide a high level of light, but not overdrive the LED. The second light uses 4 aa batteries and a series resistor selected to provide the same level of light as the first flashlight when both flashlights have new batteries installed.You can only take a 12 pack of batteries with you. The batteries would be changed when the light output reached 50% of the starting brightness.Which light would give the total longest runtime on 12 batteries?

Intuitively, I would suggest the 4-cell flashlight because it would take longer for the battery voltage to drop to the cut off point, since the starting voltage is higher.

Doing some calculations, it seems to suggest that the cells would be changed out when they reach 0.75V each, irregardless of 3 or 4 cell configuration so I'm not sure that my initial suggestion is correct. The extra voltage from the fourth cell is dropped by the resistor.

The main difficulty is figuring rate of discharge and how fast the voltage in each cell would drop to the cut off point. This is especially difficult because the current draw constantly changes according to the battery voltage. Higher voltage, higher current, faster discharge. As voltage drops, so does current draw and therefore rate of discharge.

Efficiency of the 4-cell flashlight would be lower than for a 3-cell flashlight since the resistor in the 4-cell would consume more power. Lower efficiency means reduced runtime?

This is going to require a little more thought and perhaps some modelling.

This requires a little calculus, but I think that I have a solution worked out. I'll post the details later this evening.

Hint: Four cells do have an advantage over three.