Problem 9 · 2024 Math Kangaroo
Medium
Number Theory
sum-constraintcasework
Pia writes a number in each of the 16 little circles (see picture). Numbers in neighbouring circles differ by 1. She writes the number 5 in one circle and the number 13 in another. How many different numbers does Pia write in the 16 circles?
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Answer: A — 9
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Hint 1 of 2
Stepping from circle to circle changes the number by exactly 1, so going all the way around the ring you must take as many +1 steps as -1 steps.
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Hint 2 of 2
To get from 5 up to 13 and back, the numbers have to pass through every value between 5 and 13.
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Approach: count the values forced by going up to 13 and back to 5
- Neighbouring circles differ by 1, so as you walk around the ring the value rises or falls by 1 at each step.
- Somewhere a 5 and a 13 appear, and to climb from 5 to 13 the numbers must hit every whole number 5, 6, 7, …, 13.
- That is 9 different values, and with only 16 circles you can arrange them without needing any number outside 5–13.
- So Pia writes 9 different numbers.
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