I would say, the easiest way to do what you want is use the schematic from aaroncake, which drives 10 LED's. Then use 2 of the assemblies for 20 LED's. Which would be enough for 2 per side on a decagonally shaped disc. I wouldn't suggest using a strip, you'd have to use a relay and have each strip act as an individual lamp in the schematic (so 10 strips). Unless you did two strips all the way around, one white and one blue, and only used a 2 position counter as upposed to 10.
However, you can eliminate the IC's just be using a series of flasher circuits..considering i have no clue where to buy IC's.
http://wild-bohemian.com/electronics/flasher.html
That's your better option. You can put blue on one side and white on the other. As typical, it'l run on 12 VDC is you get 12V components. You can use the 3.6 V LEDs radioshack has (or a 50 pack off ebay) and mirror what I did for hybrid- 3 blue and 3 white per 'layer', and as many layers as needed to light up the case, to the point that the 470 ohm resistors may be able to cut down to 100 ohms.
100 K resistors :
http://www.radioshack.com/product/in...entPage=search : .99
470 resistors :
http://www.radioshack.com/product/in...entPage=search : .99
10 uf control capacitors :
http://www.radioshack.com/product/in...tId=2062393&cp : 1.59
2N3904 switching transistor :
http://www.radioshack.com/product/in...tId=2062609&cp : .79