While all mathematicians are aware of the problem regarding the most efficient route for a salesman to travel in a cross-country tour, they probably do not know the one that my now-82-year-old grandfather assigned to me 4 years ago. Since the problem has a particular quirk that he, I, and others cannot seem to find published in mathematical literature, he coined it as the DeMarchis Riddle. It reads as follows:

“A salesman knocked on the door of a house, and a woman answered. He gave his pitch and asked the woman if she would like to purchase his product. The woman agreed on the condition that the salesman correctly guess the ages of her four children. She gave the following clues: ‘1. The children are all under the age of 10, and they each have different ages.    2. If you take my children’s ages in descending order, express them as a four-digit number, subtract the sum of the ages from that number, divide the resultant difference by 9 and add that quotient to the original number, you obtain my house number.’ The salesman looked at the woman’s mailbox and read 9489. In less than 30 seconds, he provided the woman with all four correct ages. What are they?”

The run-on sentence of arithmetic operations may seem daunting, but they are, after all, arithmetic operations, so there is no need for any obscure geometric modeling. Taking this problem step-by-step and wisely assigning variables, the problem is very straightforward.

Let begin organizing the problem through the following variable assignments:

x – ages in descending order expressed as 4-digit number

y – sum of the ages

The first operation mentioned by the woman is the subtraction of the age number from its sum. Clearly, that is x – y.

We can express the division of that difference by 9 as (x – y) / 9.

That quantity is then added to the original number to yield 9489: (x – y) / 9 + x = 9489

First, we make a common denominator of 9 to give (10x – y) / 9 = 9489.

Multiplying the 9 over gives 10x – y = 85401

In the end, we are interested in obtaining x, so we divide the equation above by 10: x – (y/10) = 8540.1

Now, a decimal in the right-hand side throws a curveball in an otherwise trivial problem. Keep in mind that x must be a whole number, so if x = 8540.1 + y/10, then y must have a fraction of 9/10 to add to the fraction of 1/10 in x. Here are the possibilities:

1. x = 8540.1 + 9/10 = 8541
2. x = 8540.1 + (19/10) = 8542
3. x  = 8540.1 + (29/10) = 8543
4. ….etc….

While one could perform trial-and-error by performing the relevant operations on each possible x value, he or she could also recognize that y is simply the sum of the digits of x. In the first possibility, the ages are 8, 5, 4, and 1, which sum to 18. This does not match the y value of 9. In the second possibility, y is 19, which is the sum of of the digits of 8542. Conducting test operations confirms that the women’s children are indeed 8, 5, 4, and 2 years old.

Such a basic algebra problem may seem straight from a sixth or seventh grade textbook, but there is a special anomaly that allows one to solve this problem exponentially faster. Perusing the various steps of the problem and applying your knowledge of the number 9 in particular, can you deduce such a shortcut?