Problem with making 1 bit RAM memory using transistors
Well, @Michael George... I can't refuse such a tempting promise:)
You can build the most elementary memory cell only by two inverters (transistor switches). I do this every year with my students at the beginning of the laboratory exercise about latches in the laboratory of digital electronics (you can see how in this movie and also in this Wikibooks story written by my students). Here is the building &quot;scenario&quot;.

To make a latch (flip-flop, memory cell, 1-bit RAM...), you need only a non-inverting amplifier - just connect its output to the input... and it will begin &quot;memorizing&quot;... keeping by itself at the &quot;top&quot; (+Vcc) or the &quot;bottom&quot; (ground). BTW this clever idea was proposed in 18th century by Baron Münchhausen... so that he can be considered as the inventor of the most elementary RAM cell:)
But, for some reasons, in electronics there are only inverting amplifying stages - a common-emitter amplifier, in your case. So, to make a non-inverting stage, you connect  (by the yellow wire, in the picture below) in succession (cascaded) two inverting stages, and then connect (by the black wire) the output of this non-inverting amplifier to its input. Actually the two stages are connected in a loop but for some reasons (maybe to hide the idea:) they draw this configuration as two cross-coupled stages. You should connect the base resistors R3, R4 to extend the small (0.7 V) input range of the the bipolar transistor up to Vcc (5 V). If you don't connect them, the forward-biased base-emitter junction (like a diode) will fix the collector voltage of the other transistor at +0.7 V, and your LEDs will never light (if connected to ground) or will ever light (if connected to Vcc).
You can drive this latch in various ways (as you can see in the movie):

The most correct of them is to connect the one end of the (red) wire to the ground and to touch the base of the turned-on transistor with the other end, or to touch the collector (the body, if it is made metallic) of the turned-off transistor (this is the role of your additional transistors T1 and T3, they act as the red wire).
An incorrect way is to connect the red wire to Vcc (as shown in the picture), and to touch the collector of the turn-on transistor... but it is interesting that it still survives:). It is extremely interesting to see why since exactly in this way the memory cells of today's modern RAMs are controlled...
And the most incorrect way is to touch the red wire directly to the base of the cut-off transistor (since you apply +5 V across the base-emitter junction that tries to fix it at 0.7 V). The result - more likely, either the transistor or the power supply will be damaged; that is why (to limit the current) you have to connect the base resistors R. But in the circuit shown in the movie, the transistors still survived when my students touched their bases by +5 V... it is interesting to see why...