Problem 9 · 2020 Math Kangaroo
Medium
Number Theory
factor-pairswork-backward
Five boxes contain 2, 3, 4, 7 and 15 balls. Peter wants to move balls between boxes so that every box ends up holding twice or half as many balls as one of the other boxes. At least how many balls must he move?
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Answer: A — 1
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Hint 1 of 2
Try to reach a doubling chain; notice the total 2+3+4+7+15 = 31 stays fixed.
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Hint 2 of 2
Compare {2,3,4,7,15} with a valid arrangement and count the fewest balls you must shift.
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Approach: find the nearest valid doubling arrangement
- The boxes hold 2, 3, 4, 7, 15 - total 31 balls, which stays fixed.
- A working layout (each box double or half of another) is {2, 4, 4, 7, 14}: 2=4/2, 4=2x2, 14=2x7.
- Getting there from {2,3,4,7,15} just moves one ball (from the box of 15 to the box of 3).
- So the least change needed is 1 ball.
Mark:
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