Can you beat the Strassen's Algorithm?

A Kaboobly Dooist wishes to develop a matrix-multiplication algorithm that is asymptotically faster than Strassen's algorithm .

His algorithm will use the divide-and-conquer method, dividing each matrix into pieces of size $n/4 \times n/4$ and the divide and combine steps together will take $\Theta(n^2)$ time.

If his algorithm creates $a$ subproblems, then the recurrence for the running time $T(n)$ becomes $T(n) = a\,T(n/4) + \Theta(n^2)$

What is the largest integer value of $a$ for which Professor Caesar's algorithm would be asymptotically faster than Strassen's algorithm?

Details:

• If the running time of the Strassen's algorithm is $T(n)$ , it satisfies the relation $T(n) = 7\, T(n/2) + \Theta (n^2)$

• Problem Courtesy: Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein