If you put x darts into the 15 point area and y darts into the 7 point area, you get 15x+ 7y points. So the mathematics question you are asking is "what is the largest number thatcannot[ be written 15x+ 7y for non-negative integers x and y."

7 divides into 15 twice with remainder 1 so 15(1)+ 7(-2)= 1. If A is any number then 15(A)+ 7(-2A)= A. x= A, y= -2A will give any number A. Of course, -2A is not a non-negative number. In fact, x= A- 7k and y= -2A+ 15k is also a solution for any integer k: 15(A- 7k)+ 7(-2A+ 15k)= 15A+ 7(15)k- 14A+ 7(15)k= A since the "7(15)k" terms cancel out.

So we must have and . Those are the same as and :

Well, obviously, that is always possible if there is an integer between and . In particular it is certainly possible if . That is, it is possible for any A> 105.

But that doesn't mean it isn't possible for any number less than or equal to 105. But at least we have only a finite number of cases to check:

If A= 105, and so k= 14 or 15 will work. If k=14, x= 105- 7(14)= 105- 98= 7 and y= -210+ 15(14)= -210+ 210= 0. You can get 105 points by putting 7 darts in the 15 point circle and none in the 7 point circle. If k= 15, x= 105- 7(15)= 0 while y= -201+ 15(15)= 15. You can get 105 points by putting 0 darts in the 15 point circle and 15 points in the 7 point circle.

If A= 104, [tex]= 13 and 13/15 while = 14 and 6/7. k= 14 is between them. x= 104- 7(14)= 6 and now y= -2(104)+ 14(15)= -208+ 210= 2. You can get 104 points by putting 6 darts in the 15 point circle and 2 darts in the 7 point circle.

If A= 103, = 13 and 11/15 while = 14 and 6/7. k= 14 is still between those two numbers. x= 103- 7(14)= 5 and y= -2(103)+ 14(15)= -206+ 210= 4. You can get 103 points by putting 5 darts in the 15 point circle and 4 darts in the 7 point circle.

If A= 102, = 13 and 3/5 while = 14 and 4/7. k= 14 is still between those two numbers. x= 102- 7(14)= 4 and y= -2(102)+ 14(15)= -204+ 210= 6. You can get 102 points by putting 4 darts in the 15 circle and 6 darts in the 7 point circle.

If A= 101, = 13 and 7/15 while = 14 and 3/7. k= 14 is still between those two integers. x= 101- 7(14)= 3 and y= -2(101)+ 14(15)= -202+ 210= 8. You can get 101 points by putting 3 darts in the 15 circle and 6 darts in the 7 circle.

If A= 100, = 13 and 1/3 while = 14 and 2/7. k= 14 is still between those two integers. x= 100- 7(14)= 2 and y= -2(100)+ 14(15)= -200+ 210= 10. You can get 100 points by putting 2 darts in the 15 point circle and 10 darts in the 7 point circle.

If A= 99, = 13 and 1/5 while = 14 and 1/7. k= 14 is still between those two integers. x= 99- 7(14)= 1 and y= -2(99)+ 14(15)= -198+ 210= 12. You can get 99 points by putting 1 dart in the 15 point circle and 12 darts in the 7 point circle.

If A= 98, = 13 and 1/15 while = 14. k= 14 again. x= 98- 7(14)= 0 and y= -2(98)+ 14(15)= -196+ 210= 14. You can get 98 points by putting 0 darts in the 15 point circle and 14 darts in the 7 point circle.

If A= 97, = 12 and 14/15 while = 13 and 6/7. Now k= 13 will work.

If A= 96, = 12 and 4/5 while = 13 and 5/7. Again, k= 13 will work.

If A= 95, = 12 and 2/3 while = 13 and 4/7. k= 13 will work.

If A= 94, = 12 and 8/15 while = 13 and 3/7. k= 13 will work.

If A= 93, = 12 and 2/15 while = 13 and 2/7. k= 13 will work.

If A= 92, = 12 and 4/15 while = 13 and 1/7. k= 13 will work.

If A= 91, = 12 and 2/15 while = 13. k= 13 will work.

If A= 90, while = 12 and 6/7. k= 12 will work.

If A= 89, = 11 and 13/15 while = 12 and 5/7. k= 12 will work.

If A= 88, = 11 and 2/3 while = 12 and 4/7. k= 12 will work.

If A= 87, = 11 and 3/5 while = 12 and 3/7. k= 12 will work.

If A= 86, = 11 and 7/15 while = 12 and 2/7. k= 12 will work.

If A= 85, = 11 and 1/3 while = 12 and 1/7. k= 12 will work.

If A= 84, = 11 and 1/5 while = 12. k= 12 will work.

If A= 83, = 11 and 1/15 while = 11 and 6/7.

Finally! There is no integer between 11 and 1/15 and 11 and 6/7 so there are no x and y that will make 15x+ 7y= 83.

Since every number large than 83 can be written that way, 83 is the largest number that cannot be written 15x+ 7y and so is the largest number of points that cannot be achieved by throwing darts at 15 point and 7 point circles.