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### 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/
```
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