I do not believe those lights are to signify whose move it is. You determine that by the position of the plunger. The raised plunger has some white underneath it to help visually indicate whose turn it is. The red lights are supposed to indicate when a player's flag has fallen.
chess timer help
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I have this type of chess timer and when I use it the little lights that signify whose move it is don't light up. If you have one like this, please help to find a way to turn them on? Here is a link to what it looks like, thankyou.
http://www.amazon.com/Game-Time-II-Chess-Timer/dp/B00006699U