Well, getting it open my be a bit tricky.
As I sit here going through it's different flashing types it reminds my of an octopus a squid or a cuttlefish I think it is that changes colors.

 

This is the Flashing Stick.

 

Hello,

The video is not available.

Bertus

 

Some observations: If using a rotating pattern full circle with 10 LEDs, ( 9 might be better allowing an off position ), using surface mt. LEDs, 2.8 mm wide would give a light dimater of about 13 mm.
To give a smoother rotation, we could knock off the edges with an RC of about 470 ohms & 200 to 500 uF cap on each LED. LED current around 5 mA. Better control if a LED driver were allowed.
What is the power supply ?

 

I put the flashing light stick video on You Tube, you should be able to see it now.