Problem Description

You need to convert a decimal number to a special hexadecimal format called "Hexspeak".

The conversion process works as follows:

1. First, convert the decimal number to its hexadecimal representation (using uppercase letters A-F)
2. Replace all occurrences of digit '0' with the letter 'O'
3. Replace all occurrences of digit '1' with the letter 'I'

After these replacements, the result is considered valid "Hexspeak" only if it contains exclusively letters from the set {'A', 'B', 'C', 'D', 'E', 'F', 'I', 'O'}. This means no other digits (2-9) can appear in the hexadecimal representation after the replacements.

Given a string num representing a decimal integer, you need to:

• Convert it to Hexspeak following the rules above
• Return the Hexspeak representation if it's valid
• Return "ERROR" if the result contains any characters outside the allowed set

For example:

• If the decimal number converts to hex "1A0B", after replacements it becomes "IAOB" which contains only valid letters, so it's valid Hexspeak
• If the decimal number converts to hex "3A2", after replacements it becomes "3A2" which contains the digit '3' and '2', making it invalid, so return "ERROR"

Intuition

The problem is essentially asking us to perform a series of transformations and then validate the result. Let's think through what we need to do step by step.

First, we need to convert a decimal number to hexadecimal. Python provides a built-in hex() function that does this conversion for us. The result will be in lowercase with a '0x' prefix, so we'll need to handle that.

Next, we need to replace specific digits with letters. The replacements are straightforward: '0' becomes 'O' and '1' becomes 'I'. This is a simple string replacement operation.

The key insight is recognizing what makes a valid Hexspeak string. After our replacements, the only valid characters are {'A', 'B', 'C', 'D', 'E', 'F', 'I', 'O'}. Notice that:

• Letters A-F come naturally from hexadecimal conversion
• 'I' and 'O' come from our replacements of '1' and '0'
• Any other digits 2-9 in the hexadecimal representation would make it invalid

So our validation strategy becomes: after performing all transformations, check if every character in the resulting string belongs to the valid set. If any character doesn't belong (like digits 2-9), we know it's not valid Hexspeak.

This leads us to a clean solution: convert to hex, apply replacements, then validate by checking if all characters are in our allowed set. Using Python's all() function with a set membership check gives us an elegant way to validate every character at once.

Learn more about Math patterns.

Solution Approach

The solution follows a straightforward simulation approach with these steps:

1. Define the valid character set: Create a set s containing all valid Hexspeak characters: {'A', 'B', 'C', 'D', 'E', 'F', 'I', 'O'}. Using a set allows for O(1) membership checking.

2. Convert decimal to hexadecimal:

   • First convert the input string num to an integer using int(num)
   • Apply Python's hex() function to get the hexadecimal representation
   • The hex() function returns a string like '0x1a2b', so we slice from index 2 onwards using [2:] to remove the '0x' prefix

3. Transform to Hexspeak format:

   • Convert the hexadecimal string to uppercase using .upper() since Hexspeak requires uppercase letters
   • Replace all occurrences of '0' with 'O' using .replace('0', 'O')
   • Replace all occurrences of '1' with 'I' using .replace('1', 'I')
   • These replacements can be chained together for cleaner code

4. Validate the result:

   • Use Python's all() function with a generator expression to check if every character c in the transformed string t exists in our valid set s
   • The expression all(c in s for c in t) returns True only if every character is valid
   • If valid, return the transformed string t; otherwise, return 'ERROR'

The entire solution is implemented in a single line (after defining the valid set), making it both concise and efficient. The time complexity is O(n) where n is the length of the hexadecimal representation, and the space complexity is O(1) for the fixed-size set plus O(n) for the string manipulations.

Example Walkthrough

Let's walk through the solution with the decimal number "257":

1. Define valid character set:

   • We create s = {'A', 'B', 'C', 'D', 'E', 'F', 'I', 'O'}

2. Convert decimal to hexadecimal:

   • Convert string "257" to integer: 257
   • Apply hex(257) which gives us "0x101"
   • Remove the "0x" prefix: "101"

3. Transform to Hexspeak format:

   • Convert to uppercase: "101" → "101" (no change since digits remain digits)
   • Replace '0' with 'O': "101" → "1O1"
   • Replace '1' with 'I': "1O1" → "IOI"

4. Validate the result:

   • Check each character in "IOI":
     • 'I' is in our valid set ✓
     • 'O' is in our valid set ✓
     • 'I' is in our valid set ✓
   • Since all characters are valid, return "IOI"

Now let's see an example that results in "ERROR". Consider decimal number "747":

1. Convert to hex: hex(747) gives "0x2eb", remove prefix → "2eb"

2. Transform:

   • Uppercase: "2EB"
   • Replace '0' with 'O': "2EB" (no zeros to replace)
   • Replace '1' with 'I': "2EB" (no ones to replace)

3. Validate:

   • Check '2' → NOT in valid set ✗
   • Since we found an invalid character, return "ERROR"

The key insight is that any hexadecimal digit from 2-9 will cause the validation to fail, while 0 and 1 get transformed into valid letters O and I.

Solution Implementation

Time and Space Complexity

Time Complexity: O(log n), where n is the size of the decimal number represented by num.

• Converting string to integer: O(m) where m is the length of the string num
• Converting integer to hexadecimal using hex(): O(log n) where n is the decimal value
• String operations (upper(), replace()): O(k) where k is the length of the hex string, which is O(log n)
• Checking all characters in set: O(k) where k is the length of the hex string
• Since the hex representation has length proportional to log n, the overall time complexity is O(log n)

Space Complexity: O(log n)

• Creating set s: O(1) as it contains a fixed number of characters
• Storing hex string t: O(log n) as the hexadecimal representation requires O(log n) characters
• The all() function with generator expression: O(1) additional space
• Overall space complexity is O(log n) due to the hex string storage

Learn more about how to find time and space complexity quickly.

Common Pitfalls

1. Incorrect Order of Operations with String Replacements

A common mistake is performing the replacements before converting to uppercase, or assuming the order of replacements doesn't matter.

Pitfall Example:

# Wrong approach - converting to uppercase after replacements
hex_string = hex(int(num))[2:]
hex_string = hex_string.replace('0', 'O').replace('1', 'I')
hex_string = hex_string.upper()  # This won't work correctly!

Why it fails: If you replace '0' with 'O' and '1' with 'I' before converting to uppercase, and the original hex has lowercase letters like 'a', 'b', etc., the final uppercase conversion won't affect the already-replaced 'O' and 'I', but the issue is that you might accidentally replace lowercase 'o' or 'i' if they existed in the intermediate string.

Correct Solution: Always convert to uppercase first, then perform replacements:

hex_string = hex(int(num))[2:].upper()
hex_string = hex_string.replace('0', 'O').replace('1', 'I')

2. Forgetting to Handle Large Numbers

Pitfall Example:

# Attempting to use format strings or other methods that might have limitations
hex_string = format(int(num), 'X')  # This works but be aware of potential issues

Why it matters: While Python handles arbitrarily large integers well, it's important to ensure your conversion method works for very large decimal numbers. The hex() function handles this correctly, but some alternative approaches might have limitations.

Correct Solution: Stick with hex(int(num)) which handles large numbers properly:

# This works for any size number
hex_string = hex(int(num))[2:].upper()

3. Misunderstanding Valid Character Set

Pitfall Example:

# Wrong - forgetting that after replacement, only certain characters are valid
valid_chars = set('0123456789ABCDEF')  # This includes all hex digits!

Why it fails: The problem specifically states that after replacements, only letters A-F, I, and O are valid. Digits 2-9 make the result invalid.

Correct Solution: Define the valid set as only the allowed letters after transformation:

valid_chars = set('ABCDEFIO')

4. Not Handling the '0x' Prefix Correctly

Pitfall Example:

# Forgetting to remove the '0x' prefix
hex_string = hex(int(num)).upper()  # Results in '0X...' being processed

Why it fails: The hex() function returns a string starting with '0x' (or '0X' after uppercase). If you don't remove this prefix, the '0' will be incorrectly replaced with 'O', and 'X' will be considered as part of your hexadecimal string.

Correct Solution: Always slice from index 2:

hex_string = hex(int(num))[2:].upper()