Hi Paulie,
For 18volts you need to drop 14.5 volts across the resistor to leave 3.5 volts for the LEDs (White LEDs I assume). LEDs normally need about 20 mA to light up.
Ohms law states Voltage =Current in amps multiplied by Resistance in Ohms
14.7 volts = 0.020 Amps X ?Ohms
which works out to 735 Ohms
So the resistors are not big enough for 18 volts (they work out just right for 12 volts).
The problem with the 425 ohm resistors is that you have increased the current to something like 14.5 / 425 = 0.035 A (35 mA)
Another equasion is Power = Voltage multiplied by Current
so the power in the resistor would be
14.5 X 0.035 = 0.45 of a Watt (The unit of power)
The power in the LED would be
3.5 X 0.035 = 0.12 of a watt
both of these are low power but even a 0.5 watt resistor would get very hot while dissipating 0.45 watts. Half, quarter an eighth watt resistors would not last very long (maybe a few hours).
I suggest you add another 425 ohm resistor or change to 1,000 ohms
the power would drop to 0.29watt and the LEDS would still be bright and cool.
Hope you can understand this