Analyzing Alternating Digit Growth in Numbers

In this article, we will explore a method to check if the digits of a given number alternately grow and shrink. This means that no two adjacent digits should be the same, and there shouldn’t be a sequence where a digit is followed by another larger digit and then another even larger digit, or vice versa.

Understanding the Problem

To illustrate this concept, we will consider a sample number and focus on its digits rather than the number itself. To accomplish this, we need to represent the number in a real base 10 format.

You may already note that numbers appear in base 10 for our perception, but fundamentally, they do not have an inherent base. To better understand how to manipulate these digits, we will use a list where each digit represents the count of units of that size in the numbering system.

Encoding and Decoding in APL

We will utilize a representation function in APL, also known as encode, to take our sample number and encode it in a specific radix (base). In our case, we specify that every position in the base 10 system needs 10 of those units to represent a value in the next higher unit.

10 10 10 10 10⊤3141514131415

This line will take the number 3141514131415 and output its digits as follows:

⍝ 3 1 4 1 5

It is essential to note that certain numbering systems, like how we count time, do not strictly conform to base 10. For example, there are 1000 milliseconds in a second, 60 seconds in a minute, and so on.

To automate the process without specifying the number of digits, we can use a decoding function, referred to as decode. This function inversely maps our digits back into a single number based on the base we defined.

10⊥10 10 10 10 10⊤3141514131415

This command decodes the sequence back to the original number:

⍝ 31415

The Core of the Solution

Now that we have the digits, our next step is to compare adjacent digits to ensure they alternate in growth. By using a derived reduction function, we can analyze pairs of digits. For instance, we can reduce the digits with a window of size 2 to observe their differences.

10(⊥⍣¯1)3141514131415

It generates the differences between adjacent digits:

⍝ 3 1 4 1 5 1 4 1 3 1 4 1 5

Using the results from this difference, we can employ the Signum function, which returns 1 for positive numbers and -1 for negative ones. However, simply checking for adjacent differences won’t suffice, particularly with the presence of zeros that may disrupt our analysis.

Normalizing Differences

To better manage adjacent differences and eliminate zero discrepancies, we can take the absolute values of our differences. This will help us normalize the outputs into two distinct values. Following this, we check if all these values are equal to 2, achieving an “and” reduction.

{2-/10(⊥⍣¯1)⍵}3141514131415

This calculates the differences:

⍝ 2 ¯3 3 ¯4 4 ¯3 3 ¯2 2 ¯3 3 ¯4

While this approach is valid, we recognize there is a more streamlined method we can utilize.

A More Elegant Approach

Instead of counting and checking each difference independently, we can take advantage of basic mathematical principles by multiplying adjacent differences. If adjacent numbers signify a switch in direction (one positive, the other negative), then their products will yield negative numbers.

{×2-/10(⊥⍣¯1)⍵}3141514131415

This gives us:

⍝ 1 ¯1 1 ¯1 1 ¯1 1 ¯1 1 ¯1 1

To verify this, we check whether all the products are negative through a series of tests, ensuring that adjacent differences alternate properly.

Final Optimization

The function we designed demonstrates a pattern that is perfect for tested programming. By never explicitly mentioning the argument, we streamline our solution further using an identity function applied to yield the argument directly.

F←{∧/2=|2-/×2-/10(⊥⍣¯1)⍵}

This function can be tested with various inputs:

H←∧/0>2×/2-/10(⊥⍣¯1)
H 3141514131415

The output will confirm whether it is a zigzag number:

⍝ 1

Testing further with different sequences:

H 53141514131415
⍝ 0
H 31141514131415
⍝ 0

Conclusion

In summary, we have defined a robust approach for analyzing whether the digits of a number alternate between growing and shrinking. With techniques leveraging both representation transformations and efficient mathematical operations, we can derive solutions effectively while adhering to tested programming principles.

Thank you for reading!