Petri dish frenzy

At time $=t$ , let the number of bacterias who have not replicated even once be $a_t$ . The number of bacterias who have replicated once be $b_t$ and the number of bacterias who have replicated twice be $c_t$ . We can now express the rules of division as follows :-

$a_{t+1} = a_t + b_t + c_t$

$b_{t+1} = a_t$

$c_{t+1} = b_t$

Also at $t=0$ , $a_0 = 1, b_0 = c_0 = 0$ . Using these initial conditions and the three recurrence relations above we can evaluate the values of $a_{30}, b_{30}$ and $c_{30}$ . The values turn out to be - $a_{30} = 53798080, b_{30} = 29249425$ and $c_{30} = 15902591$ . Hence, the total number of alive bacterias at $t = 30$ are $a_{30} + b_{30} +c_{30} = 98950096$ .

Note: The above computation can also be viewed as matrix multiplication, which can be used to construct an efficient algorithm for this problem.( End of Note )