Problem 24 · 2013 Math Kangaroo
Stretch
Algebra & Patterns
custom-operationspiral-pattern
“Sum change” is a procedure in which, for a set of three numbers, each number is replaced by the sum of the other two. So for instance {3, 4, 6} becomes {10, 9, 7}, and this again becomes {16, 17, 19}. Let the starting set be {1, 2, 3}. How many such sum changes are necessary until the number 2013 appears in the set?
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Answer: E — 2013 never comes up.
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Hint 1 of 2
Run the sum-change a few times and watch what kind of set you always get.
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Hint 2 of 2
The three numbers stay clustered around a single fast-growing value — check whether 2013 is ever that value.
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Approach: iterate and spot the pattern
- Starting from {1,2,3}, the sets become {3,4,5}, {7,8,9}, {15,16,17}, {31,32,33}, ... — always three consecutive integers centred near a power of 2.
- The centre values run 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048 — jumping from {1023,1024,1025} straight to {2047,2048,2049}.
- Those triples skip right over 2013, so 2013 never appears.
Mark:
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