Problem 21 · 2013 Math Kangaroo
Stretch
Algebra & Patterns
custom-operation
Starting from three numbers, the ‘addition machine’ makes three new ones by adding each pair together. For example, from {3, 4, 6} it makes {10, 9, 7}, and running it again gives {16, 17, 19}. We feed in the three numbers {20, 1, 3} and run the machine 2013 times. What is the biggest possible difference between two of the three resulting numbers?
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Answer: D — 19
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Hint 1 of 3
Don't follow the numbers themselves; watch the gaps between them.
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Hint 2 of 3
Work out the gaps in the example before and after one run of the machine and notice they are the same three gaps.
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Hint 3 of 3
If the gaps never change, the biggest gap at the end is the biggest gap you start with.
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Approach: watch the gaps between the numbers, not the numbers
- Check the example: \(\{3,4,6\}\) has gaps 1, 2, 3, and after the machine \(\{7,9,10\}\) has gaps 2, 1, 3 — the very same three gaps, just shuffled.
- So the three gaps between the numbers never change, no matter how many times you run the machine.
- Starting from \(\{20,1,3\}\) the biggest gap is \(20 - 1 = 19\), and it is still 19 after 2013 runs, which is choice D.
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