When the function is not constant

[ghostfirefox]

Let f1, f2: {0,1, ..., 24} → {0,1, ..., 24} be such functions that f1 (k) = k + 1 for k <24, f2 (k) = k for k <24 and f1 (24) = f2 (24) = 0. Let gi1, i2, ..., I am (k) = fi1 (fi2 (... fim (k) ...)) for i1, i2, ..., im∈ {1,2}. Find the largest m for which irrespective of the selection i1, i2, ..., im∈ {1,2} function gi1, i2, ..., I am is not a constant function.

 

[HallsofIvy]

You have defined $f_1$ and $f_2$ but to define $g_{iM}$ you refer to $f_{iM}$ which have NOT been defined.