Dartboard ScoringDate: 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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