Vol 2, No 10 Answers
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ANSWERS TO
by Bill Graham
Two of the functions are very powerful at taking large input numbers and producing extremely large output numbers. They are B and D, the functions that square a number, or multiply it by itself. It makes sense to save them for last. The other three functions, A, C and E, can be arranged in six ways. Intuitively, I thought that since C multiplies the input by seven, it should be saved for last. However, wanting to be sure, I tried all six possible arrangements. The one with the largest output was E first, then C, and A last. If you put the input number of two into E(x), the output is 11. That goes into C(x) producing an output of 78, which goes into A(x), which has an output of 309. At this point my intuition steered me wrong again. I thought that 3x2 should come last because it not only multiplies its input by itself, but then multiplies it by three also. However, upon more careful consideration, I realized that if you put x2 into 3x2, the final output is equal to 3x4. If you put 3x2 into x2, the final output is 9x4. My highest output number with an input of two came with the order E, C, A, B and D. The output of E was 11, then 78, then 309 as I said earlier. The input of 309 into D(x) results in multiplying 309 x 309 x 3, which equals 286,443. When that is squared by B(x), the final output is 82,049,592,249. Extra credit: 1) Suppose we want the smallest output number? Is the order of the functions necessarily reversed? 2) Suppose the input number is something other than two? Between zero and one maybe? Maybe -2? To get the biggest output, does the order of the functions always remain the same?
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There are 120 possible arrangements of these functions.
Rather than try all of them, let’s use some function strategy.