Solution:
The answer is that each nectarine costs 24 cents.
Construct a table of all combinations of six coins that uses the smallest number of coins necessary.
For example, you would not use 0 Quarters, 3 Dimes, 0 Nickels, and 3 Pennies because you could have used 1 Nickel and 1 Quarter
in place of the 3 Dimes (using 5 coins instead of 6).
The table shows the cost of a nectarine if she bought 1 nectarine, 2 nectarines, or 3 nectarines for each of the coin combinations. You need to find one
price that occurs in each of the last three columns.
| Q |
D |
N |
P |
= |
1 |
2 |
3 |
| 0 |
1 |
1 |
4 |
= |
.19 |
- |
- |
| 0 |
2 |
0 |
4 |
= |
.24 |
.12 |
.08 |
| 1 |
0 |
1 |
4 |
= |
.34 |
.17 |
- |
| 1 |
1 |
0 |
4 |
= |
.39 |
- |
.13 |
| 1 |
1 |
1 |
3 |
= |
.43 |
- |
- |
| 1 |
2 |
0 |
3 |
= |
.48 |
.24 |
.16 |
| 2 |
0 |
0 |
4 |
= |
.54 |
.27 |
.18 |
| 2 |
0 |
1 |
3 |
= |
.58 |
.29 |
- |
| 2 |
1 |
0 |
3 |
= |
.63 |
- |
.21 |
| 2 |
1 |
1 |
2 |
= |
.67 |
- |
- |
| 2 |
2 |
0 |
2 |
= |
.72 |
.36 |
.24 |
| 3 |
0 |
0 |
3 |
= |
.78 |
.39 |
.26 |
| 3 |
0 |
1 |
2 |
= |
.82 |
.41 |
- |
| 3 |
1 |
0 |
2 |
= |
.87 |
- |
.29 |
| 3 |
1 |
1 |
1 |
= |
.91 |
- |
- |
| 3 |
2 |
0 |
1 |
= |
.96 |
.48 |
.32 |
| 4 |
0 |
0 |
2 |
= |
1.02 |
.51 |
.34 |
| 4 |
0 |
1 |
1 |
= |
1.07 |
- |
- |
| 4 |
1 |
0 |
1 |
= |
1.11 |
- |
.37 |
| 4 |
1 |
1 |
0 |
= |
1.15 |
- |
- |
| 4 |
2 |
0 |
0 |
= |
1.20 |
.60 |
.40 |
| 5 |
0 |
0 |
1 |
= |
1.26 |
.63 |
.42 |
| 5 |
0 |
1 |
0 |
= |
1.30 |
.65 |
- |
| 5 |
1 |
0 |
0 |
= |
1.35 |
- |
.45 |
| 6 |
0 |
0 |
0 |
= |
1.50 |
.75 |
.50 |
As you can see from the table, only one number occurs in each of the last three columns: 24 cents.
At first, Millicent had 2 dimes and 4 pennies (24 cents) to pay for 1 nectarine.
Then she had 1 quarter, 2 dimes, and 3 pennies (48 cents) to pay for 2 nectarines.
Finally, she had 2 quarters, 2 dimes, and 2 pennies (72 cents) to pay for 3 nectarines.
Seth Cohen noticed that this continues to work beyond the scope of the puzzle:
1 nectarine = 24 cents = 2 dimes, 4 pennies
2 nectarines = 48 cents = 1 quarter, 2 dimes, 3 pennies
3 nectarines = 72 cents = 2 quarters, 2 dimes, 2 pennies
4 nectarines = 96 cents = 3 quarters, 2 dimes, 1 penny
5 nectarines = 120 cents = 4 quarters, 2 dimes
The reason this works is that each time, we add 25 cents minus 1 cent: we replace one penny with one quarter.