How to compute resistors values in bipolar transistor circuit?
Note new REFDES (..reference designators)

allowing for slight dimmer with low battery @ 8V and &lt; =20mA max at 6.2V nominal for two Blue LEDs with some wider tolerance possible depending on parts and current.

Collector R's determine the current limit say at 20mA or maybe you prefer 10mA.

Let R4 = R2= (9-6.2V nom)/ 0.02A  = 140 Ohms  (assuming Vce=0)

The "Off LEDs"   will still be dim from base current for the other Q.  

So rather than let Ib=10% of Ic we can allow the Vce to rise from 0V to &lt;= 0.5V (not fully saturated) when turned ON by requiring the current gain to be >10% of hFE or IC/Ib=10 to operate more in the linear range of Vce>>0 but not too high so it affects the current limit.

Let's choose Ic/Ib = 33

If Ib2 = 0.02A/33 = 606 uA find R1.
 Q2-C when off to bias Q1 = Vbat-dim LEDs = 9V - (2.6V*2)=3.8V with Q1-Vbe=0.6V    

R1 = 3.8V-0.6 / 606 uA = 5k3 Ohms    

R3 = (9V-0.6V)/606 uA = 14k  

These can be above or below the LEDs.

and  R2=R4=140 ohms.

If you maintain all these R Ratios and increase all R's x10  to reduce LED current  from ~ 20mA to ~ 2mA this will work.

A more clever design replaces R2 &amp; R4 with a shared R from 9V. to both diode pairs since only one pair is on.

There are other ways too using CMOS inverters.

Other info

For the student who asks the Prof (and you should) if you used 10k for each base resistor and the voltage is different how can you guarantee each LED string gets the same current when ON.  (left Rb goes 0 to Vbat=9V, right Rb from Collector,  which rises only to 3V due OFF LED leakage and voltage drop of 2.8V per Blue LED when very dim with base drive to next stage.). His answer ought to be. It works for very low currents and large collector R values.  My answer technically is more accurate and equalizes the LED brightness and current (assuming identical LEDs).

The bottom line is you are allowed to interrupt the Prof and ask a good question with your assumptions stated. DONT BE AFRAID TO ASK AND LEARN. He  will then clarify his answer and perhaps correct your assumptions or correct an oversight and error in his answer. ( I used to do this all the time with my 4th yr Semi. Prof) then he asked me to be his grad student, but I wanted to get out and design things, feed my wife and kid ...)