# 711 Price Puzzle

This week I’m going to look at the classic 7-11 math puzzle. It’s told in a variety of ways, but here is the kernel:
A girl walks into a store and picks up four items to purchase. She gets to the checkout only to find the cash register is broken. The clerk picks up a calculator and attempts to determine the sum of the four items. Unfortunately, he presses the multiplication button between the prices, not the addition button. He announces the ‘total’ to be \$7.11 The girl was watching, noticed the error, and brought it to his attention. The clerk apologized and redid the calculation, this time correctly using the addition button. “What a coincidence”, he declared, “The answer is the same, and it’s still \$7.11”

What were the prices of the four items?

## Definitions

Explicitly, if the prices of the items are: A, B, C, D then we are looking for a situation in which A+B+C+D = A×B×C×D = 7.11
It’s not stated in the puzzle, but we’re looking for answers exactly to the penny, so no rounding.
As it happens, there is only one solution to this puzzle so we have sufficient information to solve.

## Initial thinking

This is one of those puzzles for which it is tempting to write a few nested loops and brute force, rather than work it out with pen and paper, but as you might expect, badly coded, it could be terribly inefficient. With just a little thinking we can optimize our approach.
The first thing we should do is work in pennies, not decimals of dollars. Using pennies allows us to work with integers (faster, and no rounding errors). So we have just two constraints:
A+B+C+D = 711
A×B×C×D = 711,000,000
The store is not giving away product for free so, without loss of generality, even if three of the items are as cheap as possible at \$0.01 (A,B,C=1), then the highest possible price for the remaining item is \$7.08 (D=708), to fit the constraint that the total is 711. If we wanted to write a (brain dead) set of nested loops, this is the maximum we’d need to go up to (we only need three loops, not four, as the fourth price is constrained by the others)
``````
for (A = 1; A <= 708; A++) {
for (B = 1; B <= 708; B++) {
for (C = 1; C <= 708; C++) {
D = 711 - (A + B + C);
if (A * B * C * D == 711000000) {
//Do stuff
}
}
}
}
```
```
I’m not, even for a moment, suggesting you write the code this way. A quick calculation reveals that the inner part, where we perform the test, would need to be performed a staggering 7083 times which is an eye watering 354,894,912 times!
Even here there is redundancy as we’ll get symmetric/redundant solutions as we are cycling through [1 … 708] for each variable. A simple optimization is to make the inner loops carry on from a minimum of the outer variable. In this way we also get the solutions in sorted order and remove symmetric solutions (it’s like just looking at the upper triangle in a matrix).
``````
for (A = 1; A <= 708; A++) {
for (B = A; B <= 708; B++) {
for (C = B; C <= 708; C++) {
D = 711 - (A + B + C);
if (A * B * C * D == 711000000) {
//Do stuff
}
}
}
}
```
```
(We’re assuming here that the prices might not necessarily be distinct; it was not stated in the puzzle, so we’re keeping our options open. If the prices were distinct, we could use B=A+1 and C=B+1)
This reduces number of executions of the inner loop to 59,400,020. A very healthy reduction, but there’s a long way to go. It’s both a good and a bad thing about the power of computers today. Even with this terrible, terrible code, this program will execute in a couple of seconds. If we’re only going to execute this code once to get an answer, is that OK? If it’s code that is going to be run millions of times, sure, we need to optimize the hell out of it, but if all we want is an answer, and we get one in two seconds with brain-dead code, is it the correct strategy to invest an additional hour or so writing more efficient code that can execute in a couple of milliseconds? This is a philosophical question in computer science that I’ll leave unanswered today as we go further down the rabbit hole on this problem.
Looking at the code above you can see we’re wasting time processing impossible numbers, and often, when the numbers are large, we’ll see that D goes negative. For instance if A=666, B=667, C=700 then this would make D=-1322 to fit the constraint. We can make another change.
``````
for (A = 1; A <= 708; A++) {
for (B = A; B <= 708 - A; B++) {
for (C = B; C <= 708 - (A + B); C++) {
D = 711 - (A + B + C);
if (A * B * C * D == 711000000) {
//Do stuff
}
}
}
}
```
```
And now the inner calculation will execute 9,879,019 times (a fraction of a second). But there is still much more we can do.
Even still, we’re exploring the inner loops, unnecessarily, when there is no way it could be a valid solution. We know that each price has to be a factor of 711,000,000 so the only solutions we should be exploring are ones where each price is a factor (hold onto that thought for later). We could enumerate these factors beforehand and cycle through them, but we could also just put a check using modulus division before we attempt to go deeper.
``````
for (A = 1; A <= 708; A++) {
if (711000000 % A == 0) {
for (B = A; B <= 708 - A; B++) {
if (711000000 % B == 0) {
for (C = B; C <= 708 - (A + B); C++) {
if (711000000 % C == 0) {
D = 711 - (A + B + C);
if (A * B * C * D == 711000000) {
//Do stuff
}
}
}
}
}
}
}
```
```
We’re now down to 29,256 checks for the inner loop. Over four orders of magnitude improvement from the starting count of 354,894,912.

## Prime Factorization

The above logic hints at another path we can go down, and some further optimizations. We know that A,B,C,D must all be factors of 711,000,000. We can go further and find the prime factors for this product. We know that the prime factors will be distributed between the four prices (the product is the same, irrespective of which prime factor is in which solution).
Prime_Factors (711,000,000) = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5 × 5 × 5 × 5 × 5 × 79
Prime_Factors (711,000,000) = 26 × 32 × 56 × 791
One of the prices will contain the prime factor 79, without loss of generality, let’s say this is A.
Let’s take a look at what other factors could be combined with A. The maximum possible value for any price, as we know, is 708, so if A = 79n, then n must be in the range [1-8], so A must be in the set {79,158,237,316,395,474,553,632}.
There is no prime factor of 7, so we can rule n=7 (A=553). We can also rule out n=8 (A=632), because if A=632, then B+C+D would need to be 79, and would need to B×C×D=711,000,000/632 = 1,125,000. There is no way we could possibly get to this value as the cube root of 1,125,00 is over 104, which is higher than 79, the total sum of B+C+D. (A useful fact is the geometric mean of any set of positive integers is always no greater than their arithmetic mean. It’s this reason that a square is the optimal rectangular shape based on minimizing the perimeter, and the cube is the optimal volume cuboid minimizing edge dimensions).
Using similar logic we can also rule out n=6 (A=474), and n=5 (A=395), so we’re down to {79,158,237,316,395,474,553,632} or n in range [1 … 4].
At the other end of the scale we can apply the same geometric mean logic. Even if three of the prices were 708 (clearly impossible), then as 711,000,000/7083 > 2 then the absolute minimum any price could be is 3, and so a new possible max is 711-3-3-3 = 702. We’ve reduced the range, plus we’ve also applied a constraint that A must be a multiple of 79 in the range [1 … 4].
``````
for (n = 1; n <= 4; n++) {
A = 79 * n;
for (B = 3; B <= 702 - A; B++) {
if (711000000 % B == 0) {
for (C = B; C <= 702 - (A + B); C++) {
if (711000000 % C == 0) {
D = 711 - (A + B + C);
if (A * B * C * D == 711000000) {
//Do stuff
}
}
}
}
}
}
```
```
This reduces the internal test to just 5018 passes.
An alternative strategy to solve this problem is to pivot and instead of iterating through the prices, is to use a multinomial distribution approach and simply distribute the prime factors between the four buckets. There are 6 lots of the prime factor 2, so there are 9C3 ways to distribute these factors between the four prices A,B,C,D (If you have N items, and B buckets, the number of ways to distribute these N items between these buckets is N+B-1CB-1). Similarly there are only two prime factors of three, and only 10 possible combinations of buckets they can be in (This even before applying the constraint that 1 ≤ n ≤ 4, so there is no way a factor of 5 can share the same bucket as 79, and it’s either one or zero 3’s, or two, one, or zero 2’s). It’s possible to try all combinations of each prime factor in each bucket and then test if A+B+C+D==711 (we know already that the product is 711,000,000 as we’re using all the prime factors).

## Solution

If you take the time to run the code you will find that there is only one solution to the problem (ignoring trivial reflections), the four prices are:
\$1.20   \$1.25   \$1.50   \$3.16

## Expansion

There is only one solution to the \$7.11 problem. Is this the only total price for which it is posssible to find four items? Now that we have our code down to a few milliseconds there’s no reason why we can’t test hundreds of prices. Below are all the results for price totals up to \$10.00
As you can see there are many solutions. The majority are unique solutions, but there are some with two, three, four, and even five distinct solutions (such as \$8.55) for the same total price.
Not all solutions have distinct prices, for instance the solution to \$7.50 is {\$1.00, \$1.00, \$2.50, \$3.00} having two items for a dollar.

## Prime Factors

If you look at the prime factors for all these solutions, you will see that each has a least six factors of 2, and at least six factors of 5. This should be no surprise.
We are factoring numbers of the format x000000, because we are converting to pennies. Each number ends in six zeroes. The only way to get a zero is through the multiplication of the prime factors 2 and 5, and we need at least six of each of them. Once zeroes appear at the end of a number, any future product has to have, at least, the same number of terminating zeroes.

## Table of results

Total ABCD Prime Factors
\$6.44\$1.25\$1.60\$1.75\$1.842,2,2,2,2,2,2,2,5,5,5,5,5,5,7,23
\$6.51\$1.25\$1.40\$1.86\$2.002,2,2,2,2,2,3,5,5,5,5,5,5,7,31
\$6.60\$1.10\$1.50\$2.00\$2.002,2,2,2,2,2,2,2,3,5,5,5,5,5,5,5,11
\$6.63\$1.25\$1.25\$1.92\$2.212,2,2,2,2,2,3,5,5,5,5,5,5,13,17
\$6.65\$1.00\$1.75\$1.90\$2.002,2,2,2,2,2,5,5,5,5,5,5,5,7,19
\$6.72\$1.12\$1.50\$1.60\$2.502,2,2,2,2,2,2,2,2,2,2,3,5,5,5,5,5,5,7
\$6.75\$1.00\$1.50\$2.00\$2.252,2,2,2,2,2,3,3,3,5,5,5,5,5,5,5,5
\$6.75\$1.20\$1.25\$1.80\$2.502,2,2,2,2,2,3,3,3,5,5,5,5,5,5,5,5
\$6.78\$1.13\$1.25\$2.00\$2.402,2,2,2,2,2,2,3,5,5,5,5,5,5,113
\$6.80\$1.00\$1.60\$1.70\$2.502,2,2,2,2,2,2,2,2,5,5,5,5,5,5,5,17
\$6.84\$1.00\$1.44\$1.90\$2.502,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5,19
\$6.84\$1.14\$1.20\$2.00\$2.502,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5,19
\$6.86\$1.00\$1.40\$1.96\$2.502,2,2,2,2,2,2,5,5,5,5,5,5,7,7,7
\$6.89\$1.06\$1.25\$2.08\$2.502,2,2,2,2,2,5,5,5,5,5,5,13,53
\$6.93\$1.00\$1.50\$1.68\$2.752,2,2,2,2,2,3,3,5,5,5,5,5,5,7,11
\$6.93\$1.75\$1.80\$2.50\$0.882,2,2,2,2,2,3,3,5,5,5,5,5,5,7,11
\$7.02\$1.17\$1.25\$1.60\$3.002,2,2,2,2,2,2,3,3,3,5,5,5,5,5,5,13
\$7.05\$1.00\$1.20\$2.35\$2.502,2,2,2,2,2,3,5,5,5,5,5,5,5,47
\$7.07\$1.00\$1.25\$2.02\$2.802,2,2,2,2,2,5,5,5,5,5,5,7,101
\$7.07\$1.75\$2.02\$2.50\$0.802,2,2,2,2,2,5,5,5,5,5,5,7,101
\$7.08\$1.00\$1.18\$2.40\$2.502,2,2,2,2,2,2,2,3,5,5,5,5,5,5,59
\$7.11\$1.20\$1.25\$1.50\$3.162,2,2,2,2,2,3,3,5,5,5,5,5,5,79
\$7.13\$1.00\$1.15\$2.48\$2.502,2,2,2,2,2,5,5,5,5,5,5,23,31
\$7.14\$1.02\$1.12\$2.50\$2.502,2,2,2,2,2,2,3,5,5,5,5,5,5,7,17
\$7.14\$1.19\$1.25\$1.50\$3.202,2,2,2,2,2,2,3,5,5,5,5,5,5,7,17
\$7.20\$1.00\$1.20\$2.00\$3.002,2,2,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5,5
\$7.20\$1.00\$1.50\$1.50\$3.202,2,2,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5,5
\$7.20\$1.50\$2.40\$2.50\$0.802,2,2,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5,5
\$7.25\$1.45\$2.50\$2.50\$0.802,2,2,2,2,2,5,5,5,5,5,5,5,5,29
\$7.26\$1.76\$2.00\$2.75\$0.752,2,2,2,2,2,2,3,5,5,5,5,5,5,11,11
\$7.28\$1.00\$1.28\$1.75\$3.252,2,2,2,2,2,2,2,2,5,5,5,5,5,5,7,13
\$7.28\$2.00\$2.08\$2.50\$0.702,2,2,2,2,2,2,2,2,5,5,5,5,5,5,7,13
\$7.29\$1.00\$1.25\$1.80\$3.242,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5
\$7.35\$1.00\$1.05\$2.50\$2.802,2,2,2,2,2,3,5,5,5,5,5,5,5,7,7
\$7.35\$1.20\$1.25\$1.40\$3.502,2,2,2,2,2,3,5,5,5,5,5,5,5,7,7
\$7.35\$1.75\$2.40\$2.50\$0.702,2,2,2,2,2,3,5,5,5,5,5,5,5,7,7
\$7.37\$2.00\$2.20\$2.50\$0.672,2,2,2,2,2,5,5,5,5,5,5,11,67
\$7.47\$1.25\$2.00\$3.32\$0.902,2,2,2,2,2,3,3,5,5,5,5,5,5,83
\$7.50\$1.00\$1.00\$2.50\$3.002,2,2,2,2,2,2,3,5,5,5,5,5,5,5,5,5
\$7.52\$1.88\$2.50\$2.50\$0.642,2,2,2,2,2,2,2,2,2,5,5,5,5,5,5,47
\$7.56\$1.12\$1.25\$1.44\$3.752,2,2,2,2,2,2,2,3,3,3,5,5,5,5,5,5,7
\$7.56\$1.25\$1.25\$1.28\$3.782,2,2,2,2,2,2,2,3,3,3,5,5,5,5,5,5,7
\$7.56\$1.25\$1.75\$3.60\$0.962,2,2,2,2,2,2,2,3,3,3,5,5,5,5,5,5,7
\$7.56\$1.26\$2.50\$3.00\$0.802,2,2,2,2,2,2,2,3,3,3,5,5,5,5,5,5,7
\$7.62\$1.00\$1.27\$1.60\$3.752,2,2,2,2,2,2,3,5,5,5,5,5,5,127
\$7.65\$1.00\$1.00\$2.25\$3.402,2,2,2,2,2,3,3,5,5,5,5,5,5,5,17
\$7.65\$1.00\$1.20\$1.70\$3.752,2,2,2,2,2,3,3,5,5,5,5,5,5,5,17
\$7.65\$1.50\$2.00\$3.40\$0.752,2,2,2,2,2,3,3,5,5,5,5,5,5,5,17
\$7.65\$2.00\$2.50\$2.55\$0.602,2,2,2,2,2,3,3,5,5,5,5,5,5,5,17
\$7.67\$2.08\$2.50\$2.50\$0.592,2,2,2,2,2,5,5,5,5,5,5,13,59
\$7.70\$1.00\$1.00\$2.20\$3.502,2,2,2,2,2,2,5,5,5,5,5,5,5,7,11
\$7.70\$1.00\$1.25\$1.60\$3.852,2,2,2,2,2,2,5,5,5,5,5,5,5,7,11
\$7.74\$1.20\$1.25\$1.29\$4.002,2,2,2,2,2,2,3,3,5,5,5,5,5,5,43
\$7.74\$1.25\$2.25\$3.44\$0.802,2,2,2,2,2,2,3,3,5,5,5,5,5,5,43
\$7.74\$1.29\$2.50\$3.20\$0.752,2,2,2,2,2,2,3,3,5,5,5,5,5,5,43
\$7.77\$1.25\$2.22\$3.50\$0.802,2,2,2,2,2,3,5,5,5,5,5,5,7,37
\$7.79\$1.25\$2.50\$3.28\$0.762,2,2,2,2,2,5,5,5,5,5,5,19,41
\$7.80\$1.00\$1.30\$1.50\$4.002,2,2,2,2,2,2,2,3,5,5,5,5,5,5,5,13
\$7.80\$1.25\$2.60\$3.20\$0.752,2,2,2,2,2,2,2,3,5,5,5,5,5,5,5,13
\$7.82\$1.00\$2.50\$3.40\$0.922,2,2,2,2,2,2,5,5,5,5,5,5,17,23
\$7.83\$2.00\$2.25\$3.00\$0.582,2,2,2,2,2,3,3,3,5,5,5,5,5,5,29
\$7.86\$1.31\$2.00\$3.75\$0.802,2,2,2,2,2,2,3,5,5,5,5,5,5,131
\$7.92\$1.00\$2.50\$3.52\$0.902,2,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5,11
\$7.92\$1.25\$2.40\$3.52\$0.752,2,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5,11
\$7.92\$1.25\$2.75\$3.20\$0.722,2,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5,11
\$8.00\$1.00\$1.00\$2.00\$4.002,2,2,2,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5
\$8.01\$1.20\$2.50\$3.56\$0.752,2,2,2,2,2,3,3,5,5,5,5,5,5,89
\$8.03\$1.00\$2.50\$3.65\$0.882,2,2,2,2,2,5,5,5,5,5,5,11,73
\$8.10\$1.20\$2.40\$3.75\$0.752,2,2,2,2,2,2,3,3,3,3,5,5,5,5,5,5,5
\$8.10\$1.25\$2.00\$4.05\$0.802,2,2,2,2,2,2,3,3,3,3,5,5,5,5,5,5,5
\$8.10\$1.35\$2.00\$4.00\$0.752,2,2,2,2,2,2,3,3,3,3,5,5,5,5,5,5,5
\$8.10\$1.50\$3.00\$3.00\$0.602,2,2,2,2,2,2,3,3,3,3,5,5,5,5,5,5,5
\$8.10\$2.40\$2.50\$2.70\$0.502,2,2,2,2,2,2,3,3,3,3,5,5,5,5,5,5,5
\$8.12\$2.32\$2.50\$2.80\$0.502,2,2,2,2,2,2,2,5,5,5,5,5,5,7,29
\$8.16\$1.00\$2.56\$3.75\$0.852,2,2,2,2,2,2,2,2,2,3,5,5,5,5,5,5,17
\$8.19\$1.00\$2.60\$3.75\$0.842,2,2,2,2,2,3,3,5,5,5,5,5,5,7,13
\$8.19\$1.04\$1.25\$1.40\$4.502,2,2,2,2,2,3,3,5,5,5,5,5,5,7,13
\$8.22\$2.50\$2.50\$2.74\$0.482,2,2,2,2,2,2,3,5,5,5,5,5,5,137
\$8.25\$2.00\$2.75\$3.00\$0.502,2,2,2,2,2,3,5,5,5,5,5,5,5,5,11
\$8.28\$1.25\$2.50\$3.84\$0.692,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5,23
\$8.28\$1.50\$2.50\$3.68\$0.602,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5,23
\$8.28\$2.30\$2.50\$3.00\$0.482,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5,23
\$8.33\$1.25\$1.75\$4.48\$0.852,2,2,2,2,2,5,5,5,5,5,5,7,7,17
\$8.33\$1.25\$2.38\$4.00\$0.702,2,2,2,2,2,5,5,5,5,5,5,7,7,17
\$8.33\$2.50\$4.00\$0.85\$0.982,2,2,2,2,2,5,5,5,5,5,5,7,7,17
\$8.36\$1.10\$2.50\$4.00\$0.762,2,2,2,2,2,2,2,5,5,5,5,5,5,11,19
\$8.37\$1.50\$2.25\$4.00\$0.622,2,2,2,2,2,3,3,3,5,5,5,5,5,5,31
\$8.40\$1.20\$2.50\$4.00\$0.702,2,2,2,2,2,2,2,2,3,5,5,5,5,5,5,5,7
\$8.40\$2.00\$2.40\$3.50\$0.502,2,2,2,2,2,2,2,2,3,5,5,5,5,5,5,5,7
\$8.45\$1.30\$2.50\$4.00\$0.652,2,2,2,2,2,5,5,5,5,5,5,5,13,13
\$8.46\$1.00\$1.25\$1.41\$4.802,2,2,2,2,2,2,3,3,5,5,5,5,5,5,47
\$8.46\$1.25\$2.82\$3.75\$0.642,2,2,2,2,2,2,3,3,5,5,5,5,5,5,47
\$8.52\$1.42\$2.50\$4.00\$0.602,2,2,2,2,2,2,2,3,5,5,5,5,5,5,71
\$8.55\$1.00\$1.00\$1.80\$4.752,2,2,2,2,2,3,3,5,5,5,5,5,5,5,19
\$8.55\$1.00\$3.00\$3.80\$0.752,2,2,2,2,2,3,3,5,5,5,5,5,5,5,19
\$8.55\$1.50\$1.50\$4.75\$0.802,2,2,2,2,2,3,3,5,5,5,5,5,5,5,19
\$8.55\$1.90\$2.40\$3.75\$0.502,2,2,2,2,2,3,3,5,5,5,5,5,5,5,19
\$8.55\$2.00\$2.25\$3.80\$0.502,2,2,2,2,2,3,3,5,5,5,5,5,5,5,19
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