MHB When the function is not constant

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The discussion centers on the functions f1 and f2, defined on the set {0,1,...,24}, where f1(k) = k + 1 for k < 24 and f2(k) = k for k < 24, with both functions returning 0 at k = 24. The main inquiry is to determine the largest integer m such that the composite function gi1, i2, ..., im, formed by applying f1 and f2 in sequence, remains non-constant regardless of the selection of indices from {1,2}. Participants note the need for clarity in defining g_{iM}, as it references f_{iM}, which has not been explicitly defined. The conversation emphasizes the importance of understanding the behavior of these composite functions to find the maximum m. Ultimately, the challenge lies in ensuring that the composite function does not simplify to a constant value for any combination of inputs.
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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.
 
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You have defined $f_1$ and $f_2$ but to define $g_{iM}$ you refer to $f_{iM}$ which have NOT been defined.
 
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