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Taming The Sum

Problem

Show that the sum, 1plus or minus2plus or minus3plus or minus...plus or minus99=100, where plus or minus between each term can be independently set to + or minus, has at least one solution.


Solution

Consider the sum, n minus (n+1) minus (n+2) + (n+3) = 0.

In other words, we can set four consecutive integers to zero:

1 minus 2 minus 3 + 4 = 0
5 minus 6 minus 7 + 8 = 0
9 minus 10 minus 11 + 12 = 0
...
93 minus 94 minus 95 + 96 = 0

Then -97 + 98 + 99 = 100

What about 12plus or minus22plus or minus32plus or minus...plus or minus992 = 100?
Is there a solution for cubes?

Problem ID: 140 (Dec 2003)     Difficulty: 2 Star

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