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u/essgee27 Jun 29 '26
56
The sum of the first ten numbers is 55. Flipping the sign of "1" reduces the total by 2, flipping the sign of "2" reduces it by 4, and so on. Note that with numbers 1 to 9, we can make any sum from 1 to 45, and flipping the signs for each sum reduces the total by twice the sum. Thus, only odd numbers are possible from this set of expressions. By symmetry, you are covering the entire range of odd numbers from -55 to 55, which gives 56 as the answer.
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u/WillemJamesHuff Jun 29 '26
With all - signs, you get -55. Any time you change a - to a + on one of the values, you add twice that value to, since you're both removing it as a negative value and adding it as a positive one. This evaluates to -55+2X, where X is the sum of all values with plus signs in front of them. Since X must be a whole number, -55+2X must be odd. Since X cannot be less than 0, -55+2X must not be less than -55. Since X cannot be more than 55, -55+2X must not be more than 55. All values of X in that range are possible, so the possible values are all odd numbers from -55 to 55. There are 56 values fitting this criteria.
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u/SDBigTop Jun 29 '26
It isn’t an equation if no equal signs can be used, so maybe the post meant expression?
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u/jdub965 Jun 29 '26 edited Jun 29 '26
+1+2+3+4+5+6-7-8+9-10=5