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Solutions for Session 2, Part A
See solutions for Problems: A1 | A2 | A3 | A4 | A5 | A6
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Problem A1 | |
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Some possible answers: each output number is 4 more than the last; the output numbers that appear are all the even numbers that aren't multiples of 4 (starting with 6); the output number is 2 more than 4 times the input number. Also, adding one input to the following input yields half the first output.
<< back to Problem A1
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Problem A2 | |
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You can't be sure, because the pattern is not completely specified, but it would be likely that the 100th number is 402. This follows the third rule listed above -- that the output number is 2 more than 4 times the input number.
<< back to Problem A2
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Problem A3 | |
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Again, you can't be completely sure, but it would be likely that the 25th number is 102, because 2 more than 4 times 25 is 102. Following the same pattern, 1004 would not appear in the output column, since 1004 is not 2 more than 4 times any whole number.
<< back to Problem A3
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Problem A4 | |
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Example: Pick 7, and then follow the algorithm. 7 >> 21 >> 19 >> 38 >> 44 >> 30 is the output. The numbers at the end are the same as the pattern described in the table. Here's why: Pick n instead, which stands for a variable number. Follow the algorithm. n >> 3n >> 3n - 2 >> 6n - 4 >> 6n + 2 >> 4n + 2 is the output, which is the rule described in Problem A1.
<< back to Problem A4
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Problem A5 | |
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Some comments:
a. | This describes the way the table is built, but doesn't say what values the table begins with. |
b. | If the first input is n, the sum of the first input and the next is n + (n + 1) = 2n + 1. Doubling this gives 4n + 2, which is the formula for the table. |
c. | This is a good digit-based description of the table. |
d. | It's 4n + 2 again. An efficient, closed-form description. |
e. | Triple the input is 3n, then 2 more than the input is n + 2, so the sum is 4n + 2. |
f. | There's no way of being sure that 4n + 2 is the correct pattern. It is definitely not the only pattern that starts 6, 10, 14, 18, 22, 26, ... . |
g. | This is another example of a different continuation to the pattern. Because the given table ends with 26, this is a valid continuation. |
<< back to Problem A5
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Problem A6 | |
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The next number could be any number with a justified pattern. 4 is the clear choice, but so is 5, the next Fibonacci number, or 10, the next number you would get when counting in base 4, or 1, the next beat in a waltz. A formula could be found for any 4th number in that sequence.
<< back to Problem A6
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