Five holes of increasing size are cut along the edge of one face of a box as shown. The number of points scored when a marble is rolled through that hole is the number above the hole. There are three sizes of marbles: small, medium and large. The small marbles fit through any of the holes, the medium fit only through holes 3,4 and 5 and the large fit

![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-102.jpg?height=257&width=403&top_left_y=1086&top_left_x=1338)
only through hole 5 . You may choose up to 10 marbles of each size to roll and every rolled marble goes through a hole. For a score of 23 , what is the maximum number of marbles that could have been rolled?
(A) 12
(B) 13
(C) 14
(D) 15
(E) 16
Five holes of increasing size are cut along the edge of one face of a box as shown. The number of points scored when a marble is rolled through that hole is the number above the hole. There are three sizes of marbles: small, medium and large. The small marbles fit through any of the holes, the medium fit only through holes 3,4 and 5 and the large fit only through hole 5 . You may choose up

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-445.jpg?height=415&width=706&top_left_y=1026&top_left_x=1249)
to 10 marbles of each size to roll and every rolled marble goes through a hole. For a score of 23 , what is the maximum number of marbles that could have been rolled?
(A) 12
(B) 13
(C) 14
(D) 15
(E) 16

## Solution

We are looking for a maximum so we want to use lots of marbles. Let's start with 10 small ones. If they all go through hole $\# 1$, we have $23-10=13$ points to be divided between medium and large marbles. We could use 2 large and 1 medium $(5+5+3=13)$ and thus use $10+3=13$ marbles or we could use 4 medium and have one of these go through hole \#4 $(3+3+3+4=13)$ which gives 14 marbles. Alternatively, of the 10 small marbles, if 9 go through hole \#1 and 1 goes through hole \#2, we have scored 11 points. The 4 medium marbles can now go through hole \#3 giving a score of $11+3 \times 4=23$. This again gives a total of 14 marbles.

ANSWER: (C)