Any number of ways to solve this puzzle, really. The main problem is the shortage of time. Let's assume that Yeastcalc.com has the right calculations. We'll choose some numbers that will provide a bit of a buffer in the total number of cells, in case our compressed schedule doesn't allow quite as much growth as the calculator says. Also assume that "various size" flasks goes up to at least 4 l
So:
Lager yeast, 1.054 gravity you need 397 billion.
5-month old pack of 100-billion cells is 10% viability, so 10 billion to start with.
If you do a 2-liter starter and step it up to a 4-liter starter, you end up with 409 billion (amazing, huh?)
So: Make a 2-liter starter at around 1.040 right in the 4-l flask, and let it do its thing for about, oh, 14 hrs. Hopefully that'll let most of the potential growth happen, and keeps us on schedule.
Make up 2 more liters at a gravity of about 1.070 and just dump it in on top of your other starter. Assume that the original starter went from 1.040 > 1.010 or thereabouts, so if you double the volume with almost twice the sugar, you have a new, larger starter at 1.040 or so. You'll be in the ballpark anyway.
Let that run for 14 hrs. Toss it in the fridge for 6 hrs to crash, take it out 2 hrs before pitching time to warm up to the right temp. 14+14+6+2 = 36
What's my grade?
p.s. I'd just go buy more yeast or push back brewday, or plan ahead better, but that's me...