I'm stuck on how to solve this one!

Question

I'm stuck on how to solve this one!

Answer 1

Say someone has a score of $100.$ By adding two $4$s and subtracting a $7,$ you can make it $100+1.$ By subtracting two $4$s and adding $7,$ you can make it $100-1.$

So for any score that can be reached you can reach a score that's one more or one less, except when one of those subtractions would give you a negative number of either the $4$-point items or the $7$-point items. You clearly cannot reach $1,$ $2,$ or $3.$ To get from $4$ to $5$ you would add two $4$s and subtract a $7,$ but that cannot be done because if the score is $4$ then you don't have one of the $7$-point items to subtract. Similarly you can't have $6.$ It's not hard to see that you can't have $9$ or $10.$

But what we did with $100$ shows that once you have a sufficiently large number you can have one more or one less. Thus it is only some small numbers that cannot be reached.