Team Play Problems
Setup: On this Team Play we will investigate certain sequences of positive odd integers. To begin, choose an odd integer M greater than 1, which will be the master number for our sequence. Then write down the number 1, which will be the first term in our sequence. To obtain the next term, subtract the current term from M, record the number of factors of 2 in the result, and then divide out all these factors of 2. Continue this process to build the entire sequence.
For example, suppose that we chose M=13 to generate a sequence.
As instructed, we let the first term equal 1. To find the next term we
compute 13–1=12, note that there are two factors of 2 in 12, and
divide them out to obtain 3, the second term. We can indicate this process
more compactly by writing 1—>3(2), where the 2 in parentheses
means that we divided out two factors of 2. To determine the third term
we compute 13–3=10, then divide out a single 2 to obtain 5. Finally,
we calculate 13–5=8 and divide out three factors of 2 to obtain
1. At this point the sequence will clearly begin to repeat, so we don't
need to proceed further. In summary, for M=13 our sequence looks