CF 102697092 - The Telephone Call
We have a ten-digit phone number, but some of its digits are unknown. The input is a string of exactly ten characters. A digit means that position of the phone number is already known, while x means that the digit at that position could be anything from 0 through 9.
CF 102697092 - The Telephone Call
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Solve time: 44s
Verified: yes
Solution
Problem Understanding
We have a ten-digit phone number, but some of its digits are unknown. The input is a string of exactly ten characters. A digit means that position of the phone number is already known, while x means that the digit at that position could be anything from 0 through 9.
The task is to count how many complete ten-digit phone numbers are consistent with the information we have. Every area code is considered valid, so there are no additional restrictions on the first digits.
The key constraint is actually very small: the input always has exactly ten positions. That means even a direct enumeration would involve at most 10 10 possible phone numbers, which is far too many for a one-second limit. We need to avoid constructing the possible numbers individually. The answer itself can be 10 10, so a 32-bit integer is insufficient, although Python's arbitrary-precision integers handle it naturally.
The central edge case is an entirely unknown number. For example:
xxxxxxxxxx
has output
10000000000
because every one of the ten positions has ten choices, giving 10 10. A careless implementation using a 32-bit integer would overflow.
Another edge case is when every digit is already known:
1234567890
The correct output is
1
There is exactly one phone number matching the given information. An implementation that initializes the answer incorrectly, for example by multiplying only after encountering an x, can accidentally produce zero or some other value.
A mixed case such as
315xxxx8x9
contains five unknown positions, so the answer is 10 5 =100000. The known digits do not affect the number of choices because they are already fixed.
Approaches
A brute-force solution could generate every possible ten-digit phone number and check whether it agrees with the known positions. This is correct because every generated number represents one candidate, and checking all candidates guarantees that none are missed. However, when all ten positions are unknown, there are 10 10 candidates. Even if checking a candidate were reduced to a single constant-time operation, that is ten billion operations, far beyond what a one-second program can perform.
The observation that makes the problem simple is that the positions are independent. Every known digit has exactly one possible value, while every x has exactly ten possible values. If there are k unknown positions, the multiplication rule gives 10 k possible phone numbers. There is no interaction between positions that could invalidate a combination, because the problem explicitly treats every area code as valid.
So instead of constructing any phone number, we only need to count the x characters and raise ten to that count. The brute-force works because it explicitly considers every combination, but fails because the number of combinations is enormous. Counting the independent choices lets us represent all those combinations with a single exponentiation.
| Approach | Time Complexity | Space Complexity | Verdict |
|---|---|---|---|
| Brute Force | O(10 10 ⋅10) | O(10) | Too slow |
| Count unknown digits | O(10) | O(1) | Accepted |
Algorithm Walkthrough
- Read the ten-character phone number pattern.
- Count how many positions contain
x. Each such position can independently contain any of ten digits. - Compute 10 k, where k is the number of unknown positions. Known positions contribute a factor of one because their value is already fixed.
- Print the resulting number.
Why it works
For every unknown position, there are exactly ten choices, and the choice made at one position does not restrict any other position. Thus, if there are k unknown positions, the total number of valid combinations is the product of ten choices for each position, namely 10 k. Every known position contributes exactly one possible value, so it does not change the product. The algorithm computes precisely this product, which means every valid phone number is counted exactly once.
Python Solution
Pythonimport sysinput = sys.stdin.readline
s = input().strip()
unknown = s.count('x')answer = 10 ** unknown
print(answer)
The input is stripped so that the trailing newline is not considered part of the phone number. Since the phone number has exactly ten positions, count('x') directly gives the exponent required by the formula.
The expression 10 ** unknown computes the number of combinations without generating them. Python integers automatically grow as needed, so the maximum answer, 10 10, is represented exactly.
There are no indexing or boundary issues because the solution never accesses individual positions. It also does not need special handling for zero unknown positions. In that case, the exponent is zero and 10 0 =1, which is exactly the correct result.
Worked Examples
Sample 1
Input:
315xxxx8x9
There are five x characters. The state of the calculation is:
| Position | Character | Unknown count |
|---|---|---|
| 1 | 3 |
0 |
| 2 | 1 |
0 |
| 3 | 5 |
0 |
| 4 | x |
1 |
| 5 | x |
2 |
| 6 | x |
3 |
| 7 | x |
4 |
| 8 | 8 |
4 |
| 9 | x |
5 |
| 10 | 9 |
5 |
The answer is 10 5 =100000.
100000
The trace demonstrates that known digits have no effect on the number of possibilities. Only the number of unknown positions matters.
Sample 2
Input:
1234567890
There are no unknown positions:
| Position | Character | Unknown count |
|---|---|---|
| 1 | 1 |
0 |
| 2 | 2 |
0 |
| 3 | 3 |
0 |
| 4 | 4 |
0 |
| 5 | 5 |
0 |
| 6 | 6 |
0 |
| 7 | 7 |
0 |
| 8 | 8 |
0 |
| 9 | 9 |
0 |
| 10 | 0 |
0 |
Thus the answer is 10 0 =1.
1
This confirms the boundary case where the entire phone number is known. There is exactly one compatible number.
Sample 3
Input:
xxxxxxxxxx
Every position is unknown:
| Position | Character | Unknown count |
|---|---|---|
| 1 | x |
1 |
| 2 | x |
2 |
| 3 | x |
3 |
| 4 | x |
4 |
| 5 | x |
5 |
| 6 | x |
6 |
| 7 | x |
7 |
| 8 | x |
8 |
| 9 | x |
9 |
| 10 | x |
10 |
The answer is 10 10 =10000000000.
10000000000
This is the largest possible answer and demonstrates why the implementation must support integers larger than 32 bits.
Complexity Analysis
| Measure | Complexity | Explanation |
|---|---|---|
| Time | O(10) | We inspect the ten characters to count the unknown positions. |
| Space | O(1) | Only the input string and a few integer variables are stored. |
The input size is fixed at ten characters, so the algorithm performs only a constant number of operations. It is comfortably within the one-second and 256 MB limits given for the problem.
Test Cases
The official problem provides three samples: a mixed pattern, a completely known number, and a completely unknown number.
Pythonimport sysimport io
def solve(): input = sys.stdin.readline s = input().strip() return str(10 ** s.count('x'))
def run(inp: str) -> str: old_stdin = sys.stdin sys.stdin = io.StringIO(inp) try: return solve() + "\n" finally: sys.stdin = old_stdin
# Provided samplesassert run("315xxxx8x9\n") == "100000\n", "sample 1"assert run("1234567890\n") == "1\n", "sample 2"assert run("xxxxxxxxxx\n") == "10000000000\n", "sample 3"
# Custom casesassert run("0000000000\n") == "1\n", "all digits known, including zero"assert run("x234567890\n") == "10\n", "exactly one unknown position"assert run("xxxxxxxxx9\n") == "1000000000\n", "nine unknown positions"assert run("1x2x3x4x5x\n") == "100000\n", "five unknown positions"
| Test input | Expected output | What it validates |
|---|---|---|
0000000000 |
1 |
All positions known, including leading and internal zeroes |
x234567890 |
10 |
Exactly one unknown position |
xxxxxxxxx9 |
1000000000 |
Nine unknown positions and the upper range of the exponent |
1x2x3x4x5x |
100000 |
Multiple separated unknown positions |
Edge Cases
When every position is known, for example 1234567890, the algorithm counts zero unknown positions and computes 10 0 =1. It does not accidentally treat the known digits as choices, so the output is exactly 1.
When every position is unknown, the input is xxxxxxxxxx. The counter reaches ten, giving 10 10 =10000000000. No enumeration takes place, so the algorithm remains just as fast as it is for the smallest cases.
When only one position is unknown, such as x234567890, there are exactly ten valid phone numbers, one for each possible digit in the first position. The algorithm counts one x and computes 10 1 =10, handling the smallest nontrivial case directly.
A pattern containing zeroes, such as 0000000000, is also handled without special treatment. A zero is a known digit, not an unknown position, so it contributes one possibility just like any other known digit. The answer is consequently 1.
The problem also allows unknown positions at any location. A pattern such as 1x2x3x4x5x has five independent unknown positions, giving 10 5 =100000. Because the algorithm counts characters rather than relying on their positions, there is no special boundary case for the first or last character.