Question

a certain game uses play money that comes in $7 bills and $9 bills. the banker can give $23 by handing over two $7s and one $9. But the banker cant make $22 with any combination of $7's and $9's. What is the largest sum that the banker can't make??

Answer 1

Any sum that the banker can make is give by
z=9x+7y
Any sum that the banker cannot give can be described by taking one less than the sum the banker can give:
w=9x+7y-1
In either case there is no maximum sum that the banker can't make.

Answer 2

No, I can't think of a proof right now, but I think the above answer is incorrect.
Here is a counter example: The banker can give 5 $9 bills and 4 $7 bills, which totals $(45+28) = $73. The answer above states that the banker cannot make a total of 1 less than that i.e. $72. Obviously the banker can make this total by handing out 8 $9 bills.
If I can come up with a proper answer, I shall post back.

Answer 3

Ok, my formula for the "sum the banker can't make is incorrect" as per the above counter-example. But the formula for the "sum the banker can make is correct, i.e.,:
z=9x+7y, where x an y must be integers greater than or equal to zero.
Since the question doesn't specify that the sum must be a whole-number, any non-interger sum cannot be made by the banker. Since the real number line goes on forever, there is no greatest sum the banker cannot make.

Answer 4

I think you have to assume that the question means the sum must be whole numbers. My way of thinking about it I find hard to explain, but if you start with a number and subtract 7 until you reach a multiple of 9 (or 0), then the sum can be made, if it goes below 0 before reaching a multiple of 9 or 0 then the sum cannot be made. Does that make any sense?

Answer 5

My original answer was ridiculous, obviously $49 can be done with seven $7. My new answer was worked out in a brute-force way, and I came out with $47.