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May 31st, 2015, 07:29 PM   #1
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The maximum impossible score

Question:
In a dart game, only 4 points or 9 points can be scored on each dart. What is the largest score that it is NOT possible to obtain? (Assume that you have an unlimited number of darts)

My attempt:

I listed the impossible scores, as follows...

1, 2, 3, 5, 6, 7, 8, 10, 11, 14, 15, 19, 23,
s
After 23, it appears that all possible numbers can be expressed as a combination of 4's and 9's.

Is 23 the answer?

As you can see, my method doesn't allow me to know that 23 is the maximum. It only suggests 23 as an answer.

Is there a better more decisive method to solve this problem?

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May 31st, 2015, 09:19 PM   #2
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GCD(4,9) = 1 which means that there exist $x,y \in \mathbb Z$ such that $4x + 9y = 1$. In fact we have $x = -2$ and $y=1$. Thus, if we can make a number $n$ using 2 or more 4s, we can also make $n+1$.

In the light of this, does the following tell you anything?
24 = 6 x 4
25 = 4 x 4 + 9
26 = 2 x 4 + 2 x 9
27 = 0 x 4 + 3 x 9
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