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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 like 1—>3(2)—>5(1)—>1(3).

Part i: Construct sequences using the same format as in the Setup section for the master numbers M=11, 23, 29, and 37.
Part ii: Now consider the sequence for M=1123. (Do not attempt to construct this sequence!) What number comes just before the number 5 in this sequence?
Part iii: The sequence for M=73 is unusually short: 1—>9(3)—>1(6). Find a value M>100 whose sequence also has exactly two terms before it begins to repeat.
Part iv: For some values of M, such as those in the first part, the resulting sequence contains every positive odd number less than or equal to (M–1)/2. We say that such a sequence is complete. Prove that if M is composite (i.e. not prime) then the corresponding sequence will not be complete.
Part v: In each of the four sequences in the first part, determine the total number of factors of 2 that were divided out before the sequence began repeating. For example, in the Setup section we divided out 2+1+3=6 factors of 2 in all. Make a conjecture relating this total to the value of M used to generate the sequence. Then prove that your conjecture holds for all complete sequences.

Team Play Soln | Return to samples page