dartboard math problem?
A dartboard has two regions for earning points: 5 points and 7 points. What is the largest score that is impossible to earn using this dartboard?
A dartboard has two regions for earning points: 5 points and 7 points. What is the largest score that is impossible to earn using this dartboard?
Comments
It is possible to earn every number of the form:
5x is possible starting with 0
5x + 7 = 5(x + 1) + 2 starting with 7
5x + 2*7 = 5x + 14 = 5(x + 2) + 4 starting with 14
5x + 3*7 = 5x + 21 = 5(x + 4) + 1 starting with 21
5x + 4*7 = 5x + 28 = 5(x + 5) + 3 starting with 28
At 28 we now have a series of numbers with every possible remainder (0 thru 4) of a multiple of 5. So every number ≥ 28 can be earned.
Now we have to test backwards from 27 until we find a number impossible to earn.
27 = 4*5 + 7
26 = 5 + 3*7
25 = 5*5
24 = 2*5 + 2*7
23 is not possible to earn. It is the largest number not possible to earn.