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In the game of darts, you can hit each of the numbers 1 – 20. Your score is doubled by landing in the outer ring and trebled by landing in the inner ring. You score 25 by hitting the outer bull’s eye and 50 by hitting the inner bull’s eye. Sarah is practicing ahead of a match. She is a good player, so all of her throws land on the board. List all of the scores, between 1 and 180, which Sarah cannot achieve with three darts, showing how you worked this out.

Is there an easy way of getting the possible scores? The obvious ones are probably 1 and 2, since all three of her darts land on the board...

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    $\begingroup$ As an aside, this has absolutely nothing to do with probability. $\endgroup$
    – JMoravitz
    Feb 27, 2017 at 18:30
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    $\begingroup$ "Sarah is practicing... she is a good player," but which scores "Jo cannot achieve with three darts"? $\endgroup$
    – hardmath
    Feb 27, 2017 at 19:10
  • $\begingroup$ Thanks, I changed it to Sarah $\endgroup$
    – user413054
    Feb 27, 2017 at 19:18

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Start thinking about the maximum scores. Maximum single = 20, maximum doubled = 40, maximum tripled = 60

You can get any sum from 3 to 22 by going 1 to 20 on first dart plus 1 + 1 on the other two;

any score from 22 to 41 by 20 on first dart plus 1 to 20 on second plus 1 on third;

any score from 41 to 60 by 20 on first, 20 on second, plus 1 to 20 on third;

any score from 61 to 80 by 30 on first, 30 on second, and 1 to 20 on third;

(Of course all of these might take more skill than most of us have, but we're talking possibilities, not probabilities here.)

any score from 81 to 100 by 50 on first, 30 on second, 1 to 20 on third (OR combinations with 40 and 40);

any score from 101 to 120 by 50 on first, 50 on second, 1 to 20 on third (OR combinations with 60 and 40);

any score from 121 to 140 by 60 on first, 60 on second, 1 to 10 on third.

Now things get more interesting above 140. To get a large odd number, you need to use either a 25 or triple of an odd number on one dart.

60 + 60 + 3*7 = 141 60 + 60 + 2*11 = 142 60 + 50 + 3*11 = 143

I am not sure if there is a systematic way for higher numbers or if you just have to build each number.

EVEN numbers 142 to 160 can be achieved by 60 + 60 + 2n where n runs 1 to 20

Odd numbers 145 and up, and even numbers over 160, need to be investigated.

This reduces the amount of work required by about 70 percent, so go ahead and check from there. I would suggest starting by listing numbers you can build easily, like 60 + 60 + 3n, and experiment with the others.

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  • $\begingroup$ I continued and I found out that for odd numbers between 141 and 157, where I used a 60, 57 and a triple from 1-20. From 159 to 179, it was a bit harder: I used at mix of 60,60 and a triple from 1-20 and 60,50 and a triple from 1-20. I found some numbers that didn't fit in:163, 169, 173, 175 and 179. For the even numbers between 160 and 180, I used 60,60 and a number between 1-20. Here too, I found some numbers that didn't fit in: 164, 166, 172, 176 and 178. Do you think that I got this right? $\endgroup$
    – user413054
    Feb 27, 2017 at 21:18
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    $\begingroup$ You're absolutely on the right track :-) I'm not going to do all the work for you -- just go over and double-check your work. $\endgroup$
    – victoria
    Feb 27, 2017 at 22:51