This week, I showcase my effort to prove an intriguing mathematical curiosity in the DeMarchis Riddle written by my grandfather. Such a proof, as will be demonstrated, has been as elusive as the quirk itself has been fascinating for nearly four years now.

Quirk in the Riddle

As you may recall, the woman’s house number was 9489. Now take that number, and perform the following operation:

However many digits are remaining in the number, subtract it by the number consisting of the largest string of common digits such that the difference remains positive.

Surely an example is necessary to convey such convoluted language. Considering that we have 9489, what is the largest four-digit number consisting of the same digits (e.g., 1111, 2222, 3333,…) that we can subtract to keep the number positive? Clearly, it is 8888.

9489 – 8888 = 601

Now, perform the same operation on the resultant difference. To maintain positivity, one must subtract 555 from 601:

601 – 555 = 46

We keep subtracting in this manner until we are left with a single-digit number, upon which we subtract it from itself to yield 0:

46 – 44 = 2

2 – 2 = 0

What do you notice about the strings of digits we utilized (8888, 555, 44, 2)? In fact each separate digit constitutes an age of one of the children (i.e., the answer to last week’s problem: 8, 5, 4, and 2)!

That is, if you take any positive integer, subtract it by its digit sum, divide that difference by 9, and add the resultant quotient to the original number, you obtain an integer which may be used to recover the original number via the operations described above. This is precisely how the salesman from the problem was able to deduce the children’s ages so quickly, as noted in the description.

Attempt at a Proof

I begin by writing out the operation that transforms a given positive integer x into a related integer y.

(x – digitSum(x))/9 + x = y

The function digitSum is mathematically defined as an expression of modulo terms and exponential summations, which be abbreviated to the eponymous name for now.

Multiplying the equation above by 9 to obtain common denominators, I obtain the following linear equation:

10x – digitSum(x) = 9y

The main question is: Why can the output y be defined as a summation of the strings of digits of x, with the length of each string proportional to the position of that digit in x? That is, how can it be that, when one performs a digitSum subtraction, division by 9, and an addition to the original x, he or she obtains the summation of the strings of digits of x?

Surely, the answer must lie with the number 9. A less significant though interesting quirk is the fact that x – digitSum(x) is always divisible by 9. Additionally, along with 3 and higher multiples, 9 is the only integer such that the divisibility of the digit sum of a given number implies the divisibility of the number itself. For example, 441 is divisible by 9 because 4 + 4 + 1  is also divisible by 9.

Clearly, I have no direction regarding a proof, originally requested by my grandfather. What I have developed, however, is a code system incorporating this very curiosity and my two favorite aspects of computer science: binary and ASCII, the subjects of next week’s post.