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Saturday, 22 December 2012

Can you put + or - to make 123456789 total as 100 ???...........153912

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You can Start with the sequence of non-zero digits 123456789. The problem is to place plus or minus signs between them so that the result of thus described arithmetic operation will be 100. 
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We got one answer

12 + 3 - 4 + 5 + 67 + 8 + 9 = 100
and suggested there existed at least one more. I do not claim to have made an exhaustive search but there appear to be more than just two answers. One of this is

123 + 4 - 5 + 67 - 89 = 100
I am sure there it at least another one. Want to find it?
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I received the following letter from Belgium:
From: Gui et Nicole RULMONT
Date: Tue, 22 Apr 1997 17:02:44 +0200

Dear Cut-the-Knot,
First please excuse my English. I am Belgian and I am very interested by your site!
You wrote in "Fun with digits": Start with the sequence of non-zero digits 123456789. The problem is to place plus or minus signs between them so that the result of thus described arithmetic operation will be 100.
Some years ago, il found in the french magazine Science et Vie the 11 solutions:

1 + 2 + 34 - 5 + 67 - 8 + 9 = 100
12 + 3 - 4 + 5 + 67 + 8 + 9 = 100
123 - 4 - 5 - 6 - 7 + 8 - 9 = 100
123 + 4 - 5 + 67 - 89 = 100
123 + 45 - 67 + 8 - 9 = 100
123 - 45 - 67 + 89 = 100
12 - 3 - 4 + 5 - 6 + 7 + 89 = 100
12 + 3 + 4 + 5 - 6 - 7 + 89 = 100
1 + 23 - 4 + 5 + 6 + 78 - 9 = 100
1 + 23 - 4 + 56 + 7 + 8 + 9 = 100
1 + 2 + 3 - 4 + 5 + 6 + 78 + 9 = 100
If we put a "-" before 1, we have one more solution:

-1 + 2-3 + 4 + 5 + 6 + 78 + 9 = 100
Using the "." decimal separation I found another solution:

1 + 2.3 - 4 + 5 + 6.7 + 89 = 100 (solution of my own)
What about 987654321 ? There are 15 solutions, said Science et Vie:

98 - 76 + 54 + 3 + 21 = 100
9 - 8 + 76 + 54 - 32 + 1 = 100
98 + 7 + 6 - 5 - 4 - 3 + 2 - 1 = 100
98 - 7 - 6 - 5 - 4 + 3 + 21 = 100
9 - 8 + 76 - 5 + 4 + 3 + 21 = 100
98 - 7 + 6 + 5 + 4 - 3 - 2 - 1 = 100
98 + 7 - 6 + 5 - 4 + 3 - 2 - 1 = 100
98 + 7 - 6 + 5 - 4 - 3 + 2 + 1 = 100
98 - 7 + 6 + 5 - 4 + 3 - 2 + 1 = 100
98 - 7 + 6 - 5 + 4 + 3 + 2 - 1 = 100
98 + 7 - 6 - 5 + 4 + 3 - 2 - 1 = 100
98 - 7 - 6 + 5 + 4 + 3 + 2 + 1 = 100
9 + 8 + 76 + 5 + 4 - 3 + 2 - 1 = 100
9 + 8 + 76 + 5 - 4 + 3 + 2 + 1 = 100
9 - 8 + 7 + 65 - 4 + 32 - 1 = 100
Write the sign "-", three solutions:

-9 + 8 + 76 + 5-4 + 3 + 21 = 100
-9 + 8 + 7 + 65 - 4 + 32 + 1 = 100
-9-8 + 76 - 5 + 43 + 2 + 1 = 100
With the decimal point:
9 + 87.6 + 5.4 - 3 + 2 - 1 = 100 (solution of my own)
If I "shuffle" the digits there are many solutions. I found some when I was young, for example:

91 + 7.68 + 5.32 - 4 = 100
98.3 + 6.4 - 5.7 + 2 - 1 = 100
538 + 7 - 429 - 13 = 100
(8×9.125) + 37 - 6 - 4 = 100 etc etc etc ....
 
Gui et Nicole Rulmont
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