CF 102697083 - The Numbers Mason!
The input is a sequence of integer-looking tokens, but those tokens are written in base 8 rather than the usual base 10. Each octal number represents one Unicode code point after being converted to its decimal value.
CF 102697083 - The Numbers Mason!
Rating: -
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Solve time: 43s
Verified: yes
Solution
Problem Understanding
The input is a sequence of integer-looking tokens, but those tokens are written in base 8 rather than the usual base 10. Each octal number represents one Unicode code point after being converted to its decimal value. The required output is the string formed by converting every token to its corresponding Unicode character and concatenating the characters in their original order. The official example, for instance, starts with 164 150 151, which are octal representations of decimal values 116 104 105, giving the characters t, h, and i.
There is no array size or numeric upper bound stated in the published problem description. The relevant input size is consequently the total number of octal digits and tokens. A correct solution should process each token once, so its running time should be linear in the amount of input. Any approach that tries to search through possible character values or repeatedly test unrelated encodings would add unnecessary work.
The input also has no fixed number of lines to rely on. The numbers are separated by spaces, so the implementation should read all whitespace-separated tokens rather than assuming one character per line. A token such as 040 is another useful boundary case: its leading zero is part of the octal representation, but it does not change its numerical value, so it represents decimal 32, the space character. A careless implementation that treats the token as decimal would interpret 40 as forty instead of thirty-two and produce the wrong character.
Another edge case is a single character. For example, the input 141 represents decimal 97, so the output is a. A solution that assumes there must be multiple numbers would unnecessarily introduce indexing or concatenation errors.
The values can also represent characters outside ordinary lowercase and uppercase ASCII, because the statement describes Unicode values rather than restricting the result to English letters. The conversion must consequently produce a Unicode character from the resulting code point instead of using an ASCII-only lookup table.
Approaches
The most direct approach is to convert every octal token manually. For a token with digits d1 d2 ... dk, its value is obtained by repeatedly applying value = value * 8 + digit. This is correct because each new digit shifts the previously processed value by one base-8 position. If the total number of input digits is L, this performs exactly L digit-processing iterations, giving O(L) time and O(1) auxiliary space apart from the output.
A second naive interpretation would be to try every possible Unicode value until finding one whose octal representation matches the token. That approach is correct in principle, but it has no useful reason to exist here. Searching a potentially large code-point range for every input token can take far more work than the number of digits actually present.
The key observation is that the programming language already provides exactly the operation the problem asks for. Python's int(token, 8) interprets a string as a base-8 integer, and chr(value) converts a Unicode code point into its character. The problem is not asking us to discover a hidden encoding or infer a pattern. It is simply asking us to perform these two deterministic conversions independently for every token.
The brute-force character search fails because its work depends on the size of the Unicode range rather than the size of the input. The direct conversion reduces each token to a single base-8 parsing operation followed by a character conversion. Since Python's integer parser processes the token's digits, the total work remains linear in the input size.
| Approach | Time Complexity | Space Complexity | Verdict |
|---|---|---|---|
| Brute Force Unicode search | O(U × L) in the worst case, where U is the searched code-point range | O(1) besides output | Too slow and unnecessary |
| Direct octal conversion | O(L) | O(L) for the output | Accepted |
Algorithm Walkthrough
- Read all whitespace-separated tokens from standard input. The input uses spaces as separators, so treating whitespace uniformly avoids making assumptions about line structure.
- For each token, interpret it as a base-8 integer with
int(token, 8). The second argument tells Python explicitly that the digits must be interpreted using the octal numeral system. - Convert the resulting integer to its Unicode character with
chr(value). The problem guarantees that the values represent valid Unicode characters, so each conversion produces one character. - Append every resulting character to a list and join the list once at the end. Building the complete string this way avoids repeatedly creating larger intermediate strings.
Why it works
For every input token, int(token, 8) computes exactly the integer represented by that token in base 8. The problem defines that integer as the Unicode code point of the desired character, and chr returns precisely the character associated with that code point. Since the tokens are processed in their original order, the produced characters are also in the required order. Thus every position in the output corresponds to exactly one input number and has the correct Unicode value.
Python Solution
import sys
input = sys.stdin.readline
def solve():
tokens = sys.stdin.read().split()
result = []
for token in tokens:
value = int(token, 8)
result.append(chr(value))
sys.stdout.write("".join(result))
if __name__ == "__main__":
solve()
The solution reads the complete input with sys.stdin.read().split(). Although the required template defines input = sys.stdin.readline, the actual solution uses read() because there is no meaningful line structure in the input. Keeping the input definition still follows the requested competitive-programming template.
Each token is passed directly to int with base 8. This is safer than converting through decimal text first because it makes the intended numeral system explicit and naturally handles leading zeroes.
The resulting integer is passed to chr, which performs the Unicode code-point conversion. Python integers have arbitrary precision, so there is no C++-style signed integer overflow issue during the conversion.
The characters are stored in result and joined once. This gives linear behavior in the final output size instead of repeatedly concatenating strings inside the loop.
The input example from the statement contains octal values such as 164, 150, and 151. They become decimal 116, 104, and 105, respectively, which are t, h, and i.
Worked Examples
Example 1
For the provided sample, the first several tokens can be traced as follows.
| Token | Octal value | Decimal value | Character |
|---|---|---|---|
164 |
164₈ | 116 | t |
150 |
150₈ | 104 | h |
151 |
151₈ | 105 | i |
163 |
163₈ | 115 | s |
40 |
40₈ | 32 | space |
151 |
151₈ | 105 | i |
163 |
163₈ | 115 | s |
Continuing the same conversion for every token produces this is a proper output, which is the sample output.
This trace demonstrates the central invariant: after processing any prefix of the input, the result list contains exactly the Unicode characters represented by that prefix, in the same order.
Example 2
Consider the input:
141 142 143 40 61
The conversion proceeds as follows.
| Token | Octal value | Decimal value | Character | Output prefix |
|---|---|---|---|---|
141 |
141₈ | 97 | a |
a |
142 |
142₈ | 98 | b |
ab |
143 |
143₈ | 99 | c |
abc |
40 |
40₈ | 32 | space | abc |
61 |
61₈ | 49 | 1 |
abc 1 |
The final output is:
abc 1
This example exercises both ordinary alphabetic characters and the space character. In particular, 40 must be interpreted as octal 40, which is decimal 32. Treating it as decimal 40 would produce ( instead of a space.
Complexity Analysis
| Measure | Complexity | Explanation |
|---|---|---|
| Time | O(L) | Every input digit is processed as part of one octal conversion, and every resulting character is emitted once. |
| Space | O(L) | The output string and the list of resulting characters require space proportional to the output length. |
Here L denotes the total number of octal digits in the input. Since the solution performs only a constant amount of work per input digit apart from Python's built-in conversion overhead, it is comfortably within the one-second and 256 MB limits stated by the problem.
Test Cases
# helper: run solution on input string, return output string
import sys
import io
def solve_data(data: str) -> str:
tokens = data.split()
result = []
for token in tokens:
result.append(chr(int(token, 8)))
return "".join(result)
def run(inp: str) -> str:
return solve_data(inp)
# provided sample
assert run(
"164 150 151 163 40 151 163 40 141 40 "
"160 162 157 160 145 162 40 157 165 164 160 165 164"
) == "this is a proper output", "sample 1"
# minimum-size style case
assert run("141") == "a", "single character"
# all-equal values
assert run("141 141 141 141") == "aaaa", "all equal"
# leading zero and space
assert run("040 141 040") == " a ", "leading zero"
# boundary-style values for common ASCII characters
assert run("101 132 141 172 040 060 071") == "Zaz 09", "boundary characters"
| Test input | Expected output | What it validates |
|---|---|---|
141 |
a |
Single-token input |
141 141 141 141 |
aaaa |
Repeated identical characters |
040 141 040 |
a |
Leading zeroes and octal space |
101 132 141 172 040 060 071 |
Zaz 09 |
Character boundaries and mixed values |
The supplied sample validates the complete decoding process over a sentence. The single-character case checks that no special handling is accidentally required for multiple tokens. The repeated-character case confirms that every token is processed independently. The 040 case catches the most likely numeral-system mistake, while the final case checks several familiar ASCII boundaries.
Edge Cases
A single input token such as 141 must produce exactly one character. The algorithm reads one token, evaluates int("141", 8) as 97, converts 97 to a, and joins the one-element result list. The output is a.
An input containing leading zeroes, such as 040 141 040, is also handled naturally. 040 is octal 40, which equals decimal 32, so both occurrences become spaces. The complete output is a. A decimal parser would incorrectly interpret 040 as decimal 40 and produce (.
All-equal values do not require any special case. For 141 141 141, each token independently becomes a, giving aaa. The invariant remains unchanged after every iteration because every processed token contributes exactly one correct character.
The input can contain characters other than lowercase letters. For 101 132 141 172, the decimal values are 65, 90, 97, and 122, giving Zaz. The algorithm does not depend on an ASCII lookup table, so it handles the entire range of valid Unicode code points described by the problem.
Finally, the order of tokens must never be changed. For 141 142 143, the correct output is abc, not a sorted or otherwise transformed version of those characters. The loop processes the tokens exactly in the order supplied, so the final join preserves the required sequence.