Re: Programming Contest

[AlZimmerma@aol.com]

Mario,

Thank you for your interest in the Darts Contest.

> ". . . the 5 areas have values (1, 2, 4, 7, 11). Then the smallest
>  unattainable score is 27."
>  
>  I don't get it.  If the three darts hit 11, isn't the (attained) score 33?
>  
>  And isn't the smallest unattainable score = largest attainable + 1?
>  
>  And so wouldn't values (Infinity, ...) be a trivial solution to any N?
>  
>  What am I missing?

It isn't true that smallest unattainable score is exactly one more than the 
largest attainable.

In the example, you are correct that 33 is the largest attainable score.  But 
there are a few scores smaller than 33 which cannot be attained.  And 27 is 
the smallest of these.

To see this, observe that scores from 1 through 26 are demonstrably 
attainable:

    1 = 1
    2 = 2
    3 = 1 + 2
    4 = 2 + 2
    5 = 1 + 4
    6 = 2 + 4
    7 = 7
    8 = 1 + 7
    9 = 2 + 7
    10 = 1 + 2 + 7
    11 = 11
    12 = 4 + 4 + 4
    13 = 2 + 4 + 7
    14 = 7 + 7
    15 = 4 + 11
    16 = 1 + 4 + 11
    17 = 2 + 4 + 11
    18 = 7 + 11
    19 = 4 + 4 + 11
    20 = 2 + 7 + 11
    21 = 7 + 7 + 7
    22 = 11 + 11
    23 = 1 + 11 + 11
    24 = 2 + 11 + 11
    25 = 7 + 7 + 11
    26 = 4 + 11 + 11

But there's no way to attain 27.

Please let me know if this doesn't answer your question.

Al Zimmermann