Why doesn't this (adapted) circuit work with the given power voltage?
I hope you're not using this "electronic die" for anything serious &mdash; it's loaded! The way the circuit is wired, it's twice as likely to show a 2 or a 3 as any other number.

In any case, the circuit comprising D1, D2, D3, R3, ZD1 and T1 is there to create a display with 6 LEDs lit whenever the counter value is "000". However, leakage current through T1 and ZD1 can allow the LEDs to light at other times. It's just a fact of life that low-voltage zeners have high leakage currents, and it only takes a few &micro;A to turn on T1 in this application.

It's a dodgy circuit design however you look at it, and the better fix for the problem might be to increase the value of R3, to maybe 100K ohms. I would be inclined to eliminate ZD1 altogether, and instead put an ordinary silicon diode in series with T1's emitter lead. Or replace that whole mess with a proper 3-input NOR gate (CD4025).