Target problem
Number Theory
Level
3
In a game, players throw darts at a target where you score either 4 or 9 points each round. What is the highest in unattainable score?
Note:This problem was inspired by someone else who gave me a similar problem.
The answer is 23.
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1 solution
You can either get 4 points or 9 points each round.
9 points can be seen as two sets of 4 points and a single 1 point.
If the point total is a multiple of 4, it will always be possible because it can consist of only 4 point scores.
If the point total is not a multiple of 4, the remainder points, wither 1, 2 or 3 points, has to be fulfilled with the 1 point from each group of 9 points.
Any total with a remainder of 1 point above 9 can be fulfilled, because you can use scores of 4 points added to one score of 9 points.
Similarily, any total with a remainder of 2 points above 18, and any total with a remainder of 3 above 27 can be fulfilled.
You can take away one group of 4 from the 27 point total, which gives a remainder of 3 but not enough groups of 4 to provide 3 single points.
27-4=23