From Matt Enlow comes a puzzle he has seen making the rounds over the years, especially in convenience stores:
I went to my local 7-Eleven and bought four items (not gas). The sum of the four prices was $\$$7.11. A happy coincidence!
But what was even stranger was that the product of the four prices was also exactly $\$$7.11, without any rounding. (When multiplying the prices, I dropped the dollar signs so that the units worked out and then multiplied their decimal values. For example, if the prices of two items were $\$$2.00 and $\$$0.32, then their product would have been 0.64.)
What were the four prices?
$711 = 3*3*79$. It is quickly determined that more factors are needed to determine answer to the problem. Since each item can have two decimal places, additional 10s, or an equal number of 2s and 5s, can be included in the prime factorization.
$711 \cdot 10^n = 2^n \cdot 3^2\cdot 5^n \cdot 79$.
I started with multiples of $0.79 as one of the items. There are only a few feasible possibilities.
| Multiple | Item Value | Product of Remaining Three Items | Sum of Remaining Three Items |
|---|---|---|---|
| 1 | $\$$0.79 | $\$$9.00 | $\$$6.32 |
| 2 | $\$$1.58 | $\$$4.50 | $\$$5.53 |
| 3 | $\$$2.37 | $\$$3.00 | $\$$4.74 |
| 4 | $\$$3.16 | $\$$2.25 | $\$$3.95 |
| 5 | $\$$3.95 | $\$$1.80 | $\$$3.16 |
| 6 | $\$$4.74 | $\$$1.50 | $\$$2.37 |
Below are some findings. I always started with the correct product. If the sum was greater than $\$$7.11, I brought a lesser and greater valued item closer together. If the sum was less than $\$$7.11, I spread a lesser and greater valued further apart.
Many are very close to the solution.
| Item 1 | Item 2 | Item 3 | Item 4 | Total Product | Total Sum |
|---|---|---|---|---|---|
| $\$$0.79 | $\$$1.50 | $\$$2.40 | $\$$2.50 | $\$$7.11 | $\$$7.19 |
| $\$$0.79 | $\$$1.60 | $\$$2.25 | $\$$2.50 | $\$$7.11 | $\$$7.14 |
| $\$$0.79 | $\$$1.80 | $\$$2.00 | $\$$2.50 | $\$$7.11 | $\$$7.09 |
| $\$$0.79 | $\$$2.00 | $\$$2.00 | $\$$2.25 | $\$$7.11 | $\$$7.04 |
| Item 1 | Item 2 | Item 3 | Item 4 | Total Product | Total Sum |
|---|---|---|---|---|---|
| $\$$1.58 | $\$$0.75 | $\$$2.40 | $\$$2.50 | $\$$7.11 | $\$$7.23 |
| $\$$1.58 | $\$$0.8 | $\$$2.25 | $\$$2.50 | $\$$7.11 | $\$$7.13 |
| $\$$1.58 | $\$$1.00 | $\$$1.50 | $\$$3.00 | $\$$7.11 | $\$$7.08 |
| Item 1 | Item 2 | Item 3 | Item 4 | Total Product | Total Sum |
|---|---|---|---|---|---|
| $\$$2.37 | $\$$0.75 | $\$$1.60 | $\$$2.50 | $\$$7.11 | $\$$7.22 |
| $\$$2.37 | $\$$0.80 | $\$$1.50 | $\$$2.50 | $\$$7.11 | $\$$7.17 |
| $\$$2.37 | $\$$0.96 | $\$$1.25 | $\$$2.50 | $\$$7.11 | $\$$7.08 |
| $\$$2.37 | $\$$1.00 | $\$$1.20 | $\$$2.50 | $\$$7.11 | $\$$7.07 |
| $\$$2.37 | $\$$1.00 | $\$$1.25 | $\$$2.40 | $\$$7.11 | $\$$7.02 |
| Item 1 | Item 2 | Item 3 | Item 4 | Total Product | Total Sum |
|---|---|---|---|---|---|
| $\$$3.16 | $\$$1.00 | $\$$1.25 | $\$$1.80 | $\$$7.11 | $\$$7.21 |
| $\$$3.16 | $\$$1.00 | $\$$1.50 | $\$$1.50 | $\$$7.11 | $\$$7.16 |
| $\$$3.16 | $\$$1.20 | $\$$1.25 | $\$$1.50 | $\$$7.11 | $\$$7.11 |
$\$$1.20, $\$$1.25, $\$$1.50, and $\$$3.16.