Find the total numbers to make A=21

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The discussion focuses on the mathematical problem of determining how to fill in blanks with "+" or "-" signs to achieve a total of A=21 using the numbers 1 through 9. It is established that A cannot equal 12 due to the constraints of the operations and the range of numbers available. The participants provide various strategies and calculations to demonstrate how to reach the total of 21 effectively.

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$A=$$\square$1 $\square$2 $\square$3$\square$4 $\square$5$\square$6 $\square$7 $\square$8 $\square$9
randomly fill in each blank with eather $"+"$ or $"-"$ ,
(1) prove $A$ can not be $12$
(2) find the total numbers to make $A=21$
 
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Albert said:
$A=$$\square$1 $\square$2 $\square$3$\square$4 $\square$5$\square$6 $\square$7 $\square$8 $\square$9
randomly fill in each blank with eather $"+"$ or $"-"$ ,
(1) prove $A$ can not be $12$
(2) find the total numbers to make $A=21$

If there is + before each number we have sum of numbers = 45
when we convert a + to a - say before n we subtract the value by 2n, hence the number shall remain odd
so $A$ cannot be 12.
we need to make the result 21 so subtract 24 so we need to choose numbers whose sum is 12 and change plus to -
the numbers are (9,3), (9,2,1), (8,4), (8,3,1), (7,5) , (7,4,1), (7,3,2), (6,5,1), (6,4,2),(6,3,2,1), (5,4,3) (5,4,2,1) that is 12 combinations.
 

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