A game awards either $7$ points or $4$ points at a time. How many possible total scores can never be reached if the game is played indefinitely?

Question

In the Martian game of QZX. a JBL is worth 7 points and a KMD is worth 4 points. There is no other way to score. Games can continue indefinitely. How many positive integer scores can never be reached? (e.g., no team could ever have a total score of 2 points.)

A) 5 B) 7 C) 9 D) 11 E) Infinity many

I don't understand this game. Can anyone explain the rules of Martian game, please?

Answer 1

There is no way a score of $1,2,3,5,6,9,10$ can be made from just $4$s and $7$s, so straight away we have $7$ scores that cannot be made, which rules out answer A. Scores of $11=4+7$ and $12=3\times4$ can be made.

Once we have reached $20=5\times4$ we can:

a) replace the five $4$s with three sevens to reach $21=3\times7$.

b) now we can replace a $7$ with two $4$s to reach $22 = 2\times7 + 2\times4$.

c) and finally we can replace another $7$ with two $4$s to reach $23 = 7 +4\times4$.

Then we have $24=6\times4$, and we can repeat the steps a), b) and c) to reach $25,26,27,\dots$. So any score from $20$ upwards can be made. This rules out answer E.

To decide between answers B, C and D, you just have to work out how many scores between $13$ and $19$ can be made with just $4$s and $7$s.

Answer 2

All integers which are multiples of $4$ are possible scores, using only KMDs.

All integers which are $1$ less than a multiple of $4$ require at least one JBL, and all these are possible using one JBL and an appropriate number of KMDs except for $3$.

All integers which are $2$ less than a multiple of $4$ require at least two JBLs to make, and all these are possible using exactly two JBLs and an appropriate number of KMDs except for $2$, $6$ and $10$.

All integers which are $3$ less than a multiple of $4$ require at least three JBLS, and all can be made using exactly three except $1,5,9,13$ and $17$. So there are nine impossible scores in total.