Dartboard Scoring

Date: 03/01/99 at 17:42:33
From: Meaghan Myers
Subject: Dartboard Scoring

What is the highest score you CANNOT get on a dart board using as many 
darts as you wish? The problem has two parts; the center is worth nine 
points and the outer ring is worth four.

Date: 03/03/99 at 10:51:43
From: Doctor Dennis
Subject: Re: Dartboard Scoring

This is a very tricky question; it took me a quite a while to figure 
out how to do it. I tried several approaches, but the way that 
eventually worked was this. First, I tried to get certain numbers by 
adding only 9's and 4's. To do this, I first tried to divide the number 
by 4, then subtracted 9 and tried to divide by 4 again, etc. until I 
either found an answer or got less than 9. Example:

19:    19 Not divisible by 4
       19-9 = 10
       10 Not divisible by 4
       10-9 = 1
       1 Not divisible by 4
       Can't subtract anymore

So 19 doesn't work.

33:    33 Not divisible by 4
       33-9 = 24
       24 divisible by 4

So 33 works.

The numbers I tried to construct were 36 (as a guess); then I moved 
down and tried 35, 34, and 33. I found that 36, 35, 34, and 33 all 
work. Now, if 4 numbers in a row work, then all numbers above them work 
just by adding 4 to each repeatedly. Example:

33 = 9 + 4 + 4 + 4 + 4 + 4 + 4
34 = 9 + 9 + 4 + 4 + 4 + 4
35 = 9 + 9 + 9 + 4 + 4
36 = 9 + 9 + 9 + 9

But then:

37 = 9 + 4 + 4 + 4 + 4 + 4 + 4 + 4 = 33 + 4
38 = 9 + 9 + 4 + 4 + 4 + 4 + 4 = 34 + 4
39 = 9 + 9 + 9 + 4 + 4 + 4 = 35 + 4
40 = 9 + 9 + 9 + 9 + 4 = 36 + 4

And then:

41 = 9 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 = 37 + 4
42 = 9 + 9 + 4 + 4 + 4 + 4 + 4 + 4 = 38 + 4
43 = 9 + 9 + 9 + 4 + 4 + 4 + 4 = 39 + 4
44 = 9 + 9 + 9 + 9 + 4 + 4 = 40 + 4

And so on forever. So, you know how to figure out if any particular 
number on the dartboard can be gotten. You also know that if you can 
get any 4 numbers in a row, then you can get all the numbers higher 
than them. So, the highest number you cannot get is the number right 
before the first series of four consecutive numbers that you can get. 

Now you should be able to figure it out.

- Doctor Dennis, The Math Forum
  http://mathforum.org/dr.math/