Well...
I tried w/o resistor and the LED's popped rather quickly, generating quite a lot of heat!
I removed the power, inserted a 100 ohm resistor and the remaining 5 (from 24!) LED's are working nicely, not getting warm at all
Told you

If you work out how the clusters are arranged, that might help determine the "ideal" resistor size. 100ohm is probably fine for just a couple strings, but you'll need to lower it if you have all of them working. Assuming 9 strings of 3 LEDs (probably right for 24 super bright white LEDs meant to be powered from 3x AAA batteries), you'll want about 20mA on each string which is 180mA total. Voltage drop will be about 3-3.5V. (5-3.3)/.18 = 9.44ohms or about a 10 ohm resistor. Maybe go 12-15ohms to be safe (it'll just dim the LEDs a bit).
I was thinking your clusters were smaller and playing it safe with the 50 ohm guess.
You've only got two strings working right now, so that's about 40mA. (5-3.3)/.04 = ~42ohms, so you might still have some room to drop it and get them brighter.
FYI, power dissipation in that 10ohm resistor is going to be .18^2*10 = 0.324W, so you'll probably want a 1W resistor. 1/2W rated is bare minimum as you usually want the rating to be 2x the expected dissipation. If you can only find 1/2W, you can put two 22ohm resistors in parallel to get 11ohms with 1W handling. Even still, it'll get toasty, so be careful to keep it away from things that don't like heat.
Note that due to the design of these stupid things, one failure tends to kill the whole device as all that current tries to flood through the remaining strings (which destroys them and forces it through what remains causing a chain reaction) which explains why the flashlights based upon these things tend to not last very long. All this so that they can save a few 3 cent resistors...