Thursday, July 7, 2011

Summer Problem Solving Marathon Solution #28

Working from right to left, we find that the first place with an obvious error is 4 + 2 = 1 in the 10,000 place. Assuming the previous parts are correct, we should have carried 1 from the previous place. So we have two possibilities:

(1) The 1 is correct, and the sum should be 11. Then either 4 needs to be replaced by 8, or 2 needs to be replaced by 6. However, replacing 4 with 8 creates an error in the hundreds place, so this possibility can be ignored.

(2) The 1 is incorrect. Then the 4 and 2 are correct, so 1 needs to be replaced with 7. But replacing 1 with 7 creates an error in the tens place.

So the only option is that 2 is replaced with 6. Making this substitution throughout, we find that the result is a correct addition. So the answer is 2 + 6 - 8.

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