Math problem about four coins

[Dragon0228]

Hi-my daughter has a math problem I cannot figure out. It is for 6th grade math from a Math mate sheet. It reads:
Just four coins, the $a, $b, $c and $d are used to exchange money in Mathland. Each coin has a whole number value and a < b < c < d. Find the value of a, b, c, and d such that all amounts of money $1, $2, $3, ... $n can be exchanged using one or two coins and n is as large as possible.

I can figure it out with small numbers, but not larger ones sinc you can only exchange each number for 1 or 2 coins. Any help would be greatly appreciated. Thank you!!

 

[mmm4444bot]

What is the largest n that you reached?

Part of this exercise is knowing when you've reached the largest dollar amount that can be exchanged using one or two coins (i.e., determining the largest value of n possible).

There are six combinations of two coins taken from a set of four.

If the set is {A, B, C, D}, then the six combinations are:

AB AC AD BC BD CD

So, for any set of four different coins that one might come up with, only 10 dollar amounts can be exchanged (not necessarily in this order):

A, B, C, D, A+B, A+C, A+D, B+C, B+D, C+D .

I asked a nephew's kid to try this exercise; here's what he came up with.

If the four coins are $1, $2, $5, and $8, then exchanging all of the amounts from $1 through $10 is possible.

1 = 1
2 = 2
3 = 1 + 2
4 = 2 + 2
5 = 5
6 = 5 + 1
7 = 5 + 2
8 = 8
9 = 8 + 1
10 = 8 + 2

Using these four coins, he also said that we could exchange $13 (8 + 5), but the highest possible n is 10 because the instructions state ALL whole numbers from 1 through n, and there's no combination for $11 and $12 using these four coins.

WHOOPS!

His overall reasoning is good, but he made a mistake. Do you see it? (Hint: $A < $B < $C < $D)

What combinations did your daughter come up with?