12V LED strips in series with a 24V power supply?

Well, yes and no. On the one hand, connecting 2 in series ought to work. On the other hand, you will need 24 volts at 36 watts.

Since each string takes 12 volts at 18 watts, its current must be 1.5 amps. Connecting them in series will still take 1.5 amps, but the total power will be 1.5 x 24, or 36.

This is an old post, but i just wanted to contribute my observation. Both your parallel and series would work, however that is if we assume that none of the LEDs would ever burn out.

If half of the leds on the 24V would have burned out on the left side (see image).

The result would be catastrophic because the remainder leds on the burned side would receive an increase in voltage (about 16 V) and on the right side the voltage would drop to about 8V.

In conclusion, as the led burn out the voltage increases on the same parallel side and if your LEDs are rated at 12V it would be just a matter of time until all lights in the same parallel burn out due to high voltage.

That is why it is recommended to always connect led strips in parallel.