I just saw a great and simple project that does just this:
http://www.youtube.com/watch?list=PLFA57ACAC0F0DE0D1&feature=player_detailpage&v=cwiLQWJq2LQ
The project is by Alan Yates:
http://www.vk2zay.net/
As I understand it, it uses a high impedance gate of a JFET to detect fluctuations in the E-field in the wires due to noise on the mains. The signal is amplified using a BJT to make sound on a piezoelectric speaker. If a light is burned out it the E-field will exist on the wire going into the light, but, not on its exit wire. Using this principal it is easy to locate the burned light. He applies this to incandescent light string, but, the same principal would apply to an LED string.
The answer is "it depends".
If you want a flashlight, with a nice beam, then you need optics, thus it is much easier to use one high-power LED so only have to purchase one optic (lens or reflector).
If you want diffused light, then having many light sources (many LEDs) is easier to work with.
Now, if you use batteries, you'll be interested in the efficiency of your LEDs (ie, lumens/W) to maximize battery life. So you need to be aware of that. There is a compromise though, as LEDs with pleasant warm light and good color rendition tend to be less efficient. Very high lumens/W LEDs are usually "cool white" and low-CRI which isn't that good for stuff like reading or ambient light.
As well, are LED strips generally brighter than the individual LEDs, which are 15,000mCD each?
mcd (tmillicandela) has nothing to do with light output. A candela is a lumen per steradian, the latter being a unit of ANGLE. This means the same LED chip, which puts out the same amount of light (lumens) can be 100 mcd or 10000 mcd depending on how the optic in front concentrates the light into a wide or tight beam. If you want to make a lamp for diffuse ambient light, you need low-mcd, high lumen LEDs.
Each individual LED is, say, 3.3v and draws 20mA. for 20, that's 1.3Whrs.
W is power, not Whr which is energy.
Now, DO NOT wires your LEDs in parallel. Since all LEDs have a bit different voltage drops, the one with the lowest voltage drop will hog all the current, then burn. Then the next one will hog the current... etc.
If you got many LEDs, you need to put them in strings with resistors to equalize the current. Or use only 1-3 high power LED.
Losing less power on the resistor means putting more LEDs in series (ie, using higher voltage) but if you start from one 3.6V LiIon cell, boosting above 12V will also be less efficient. So 12V is OK.
12V LED strips will lose some power on the resistor. 3 high power LEDs in series will not (but may be more cumbersome to use, your choice).
Best Answer
The LEDs must be connected in series strings of LEDs. A string of 8 LEDs would work, with a single small resistor to linit the current. Connect ten strings in parallel, and you have 80 LEDs.
Alternatively, connect eight strings of ten LEDs, and rely on current limiting in the power supply. This will work if the LEDs in the strings are all matched.
For the 4.5V version, connect all the LEDs in parallel, with a single current limiting resistor. Again, all the LEDs need to be reasonably well matched.