Mark:
ARROWS
scores by arrow:
Archer: 5 6 8
Bowman: 3 10 4
cumulative scores:
Archer: 5 11 19
Bowman: 3 13 17
Mark scores a bull's eye!
Faced with the same problem how could we come up with the same answer? Well one way would be to
First off, write down the facts that we know. Each score must be different, there are 6 scores, and they are in between 2 and 10. So we know that the possible scores are: 2, 3, 4, 5, 6, 7, 8, 9, 10
We also know that each cumulative score is a prime number. So here are the possible cumulative scores:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
We also know that the cumulative scores should follow this pattern:
Shot -- Archer -- Bowman
1 ------ Higher -- Lower
2 -------Lower -- Higher
3 -------Higher -- Lower
We know that Archer's first throw will be higher than Bowman's, and that both of them will shoot a prime number. Therefore, we can divide up the possible first shots into scenario's. The only possible numbers from the first shot are 2, 3, 5, 7
Possible first shot scores:
Archer: 7 Bowman: 5, 3, 2
Archer: 5 Bowman: 3, 2
Archer: 3 Bowman: 2
I don't think there's a mathematical formula to work this one out. Just trial and error, so here we will go through each possible scenario:
Scenario 1
Archer: 7 -- Bowman: 5
Now we know that Bowman must have a higher score after the next shot. The next lowest possible number he could be is 13, because Archer would have to be on 11. And Bowman cannot be any higher either since 17 minus 5 is 12 and a score of 12 in one shot is not possible. So we need to get the scores to Archer: 11 -- Bowman: 13
So Bowman can score 8 and Archer can score 4.
Archer: 11 -- Bowman: 13
Now Archer needs to be ahead, so Bowman could move onto 17 and Archer to 19. Not possible, because Archer needs to score 8 to get to 19, and an 8 has already been hit by Bowman previously. Therefore, this scenario will not work.
Scenario 2
Archer: 7 -- Bowman: 3
The only possible cumulative scores after the second round will be Archer on 11 and Bowman on 13. So Bowman scores 10 and Archer scores 4.
Archer: 11 -- Bowman: 13
The only possible cumulative scores after the third round will be Archer on 19 and Bowman on 17. Not possible because Bowman needs to score 4 and that has already been shot by Archer previously. So scenario 2 will not work.
Scenario 3
Archer: 7 -- Bowman: 2
Bowman would have to be on 13 to be ahead of Archer whose minimum score to be on is 11. This is not possible, as Bowman would need to score 11 to reach 13, and a score of 11 in one shot is not possible.
Scenario 4
Archer: 5 -- Bowman: 3
Possible scores after second throw could be Archer on 7 and Bowman on 11. It could also be 13 - 7 in Bowman's favor or 13 - 11 in Bowman's favor. So this scenario can be divided up into 3 parts:
1: Archer on 7 and Bowman on 11. This means Archer scores 2 and Bowman scores 8. Now the only possible score after the next shots is Archer on 17 and Bowman on 13. This isn't possible as Bowman would need to score a 2 and this has already been hit by Archer previously.
2: Archer on 7 and Bowman on 13. This means Archer scores 2 and Bowman scores 10. Now the only possible score after the next shots is Archer on 19 and Bowman on 17. This is not possible because Archer would have to throw 12 to get to 19 and a score of 12 from one shot is not possible.
3: Archer on 11 and Bowman on 13. This means Archer scores 6 and Bowman scores 10. Now the only possible score after the next shots is Archer on 19 and Bowman on 17. So Archer needs to score 8 and Bowman needs to score 4. The good news here is that these scores have not been scored before and so are possible, and thus we have our final answer worked out. There are other scenario's to try out, but we don't need to bother anymore as we have found our answer already.
So this is the final answer:
Bowman scored a 3, then a 10, then a 4.
Archer scored a 5, then a 6, then 8.
Cumulative scores:
Archer: 5 -> 11 -> 19
Bowman: 3 -> 13 -> 17
Archer is higher, lower, higher than Bowman, which is correct. The six dart scores are each a unique number so all requirements of the original question have been satisfied.
May I take this opportunity to wish one and all, a very happy Thanksgiving.
Don't worry about me; I'll grab a hotdog on my way to the shelter. :wink:
