Problem 14 · 2015 Math Kangaroo
Medium
Number Theory
sum-constraintcasework
Each ? in the equation \(2\,?\,0\,?\,1\,?\,5\,?\,2\,?\,0\,?\,1\,?\,5\,?\,2\,?\,0\,?\,1\,?\,5 = 0\) should be replaced by either “+” or “−” so that the equation is correct. What is the smallest number of ? that can be replaced by “+”?
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Answer: B — 2
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Hint 1 of 2
With every sign a minus, the signed total is some negative value; switching a sign to plus before a number raises the total by twice that number.
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Hint 2 of 2
You need the total to reach 0, so use the fewest plus signs whose doubled values close the gap (favour the big numbers).
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Approach: balance the signed sum to zero
- The numbers used are 2, 0, 1, 5 repeated three times; the 0s do not matter.
- Switching a sign from − to + before a number raises the total by twice that number, so the plus signs must supply exactly the gap up to 0.
- Using the large numbers first, two plus signs can close the gap while one plus sign cannot.
- So the smallest number of plus signs is 2 (B).
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