CF 162A - Pentagonal numbers

CF 162A - Pentagonal numbers

Rating: 1100
Tags: *special, implementation
Solve time: 3m 10s
Verified: yes

Solution

Problem Understanding

We need to compute the n-th pentagonal number. Pentagonal numbers form a sequence generated by a direct mathematical formula:

$$P_n = \frac{3n^2 - n}{2}$$

The input contains a single integer n, and the output is the value of this formula for that position in the sequence.

The constraints are extremely small. Since n is at most 100, even a very inefficient solution would run instantly. A loop with a few hundred operations is trivial within a 3 second limit. This also means we do not need advanced optimization techniques, precomputation, or special data structures.

The only real risk is implementing the formula incorrectly. A common mistake is misunderstanding operator precedence or forgetting the division by 2. For example, if someone writes:

3 * n * n - n / 2

instead of:

(3 * n * n - n) // 2

the result becomes wrong because only n is divided by 2.

Consider the input:

2

The correct computation is:

$$\frac{3 \cdot 2^2 - 2}{2} = \frac{12 - 2}{2} = 5$$

A careless implementation could produce 11 or 10 depending on the misplaced parentheses.

Another subtle issue is using floating-point division. Pentagonal numbers are always integers, so integer arithmetic is the cleanest approach. Using / in Python creates a float, which is unnecessary and can introduce formatting problems. Using // keeps the answer as an integer.

For example:

1

Correct output:

1

Using floating-point division might print 1.0, which would be rejected.

Approaches

The most straightforward approach is to generate pentagonal numbers one by one. The difference between consecutive pentagonal numbers grows linearly:

$$P_n - P_{n-1} = 3n - 2$$

Starting from 1, we could repeatedly add these differences until reaching the n-th value. This works because the sequence is defined incrementally. With n ≤ 100, this approach easily fits within the limits since it performs only about one hundred iterations.

The weakness of this approach is that it ignores the fact that the exact value is already given by a direct formula. Even though the iterative method is fast enough here, it performs unnecessary work.

The key observation is that the problem literally provides the closed-form expression for the answer. Since the formula directly computes the n-th value, we can evaluate it immediately in constant time.

The brute-force method works because the sequence is short, but the formula lets us skip all intermediate values and jump directly to the answer.

Approach	Time Complexity	Space Complexity	Verdict
Brute Force	O(n)	O(1)	Accepted
Optimal	O(1)	O(1)	Accepted

Algorithm Walkthrough

1. Read the integer n from input.
2. Compute the value:

$$\frac{3n^2 - n}{2}$$

We use integer arithmetic because pentagonal numbers are always integers.

1. Print the computed result.

Why it works

The formula for pentagonal numbers is mathematically defined as:

$$P_n = \frac{3n^2 - n}{2}$$

The algorithm evaluates exactly this expression for the given n. Since the implementation directly matches the definition of the sequence, the produced value is guaranteed to be the correct n-th pentagonal number.

Python Solution

import sys
input = sys.stdin.readline

n = int(input())

answer = (3 * n * n - n) // 2

print(answer)

The program begins by reading the integer n.

The main computation uses the exact mathematical definition of the pentagonal sequence. The expression:

(3 * n * n - n) // 2

matches:

$$\frac{3n^2 - n}{2}$$

The parentheses are important because the subtraction must happen before division. Without them, Python would divide only the final n by 2, changing the meaning of the formula.

The solution uses // instead of / so the result remains an integer. Pentagonal numbers are guaranteed to be whole numbers, so integer division is the correct operation.

Since the computation involves only a few arithmetic operations, the implementation runs instantly.

Worked Examples

Example 1

Input:

2

Step	Value
n	2
3 × n × n	12
3 × n × n - n	10
Final answer	5

The computation follows the formula exactly. After subtracting n from 3n², dividing by 2 gives the second pentagonal number, 5.

Example 2

Input:

5

Step	Value
n	5
3 × n × n	75
3 × n × n - n	70
Final answer	35

This trace demonstrates that the algorithm scales naturally for larger values. No loops or recursion are needed because the formula directly computes the answer.

Complexity Analysis

Measure	Complexity	Explanation
Time	O(1)	Only a fixed number of arithmetic operations are performed
Space	O(1)	The algorithm stores only a few integer variables

The constraints are tiny, so this solution easily fits within the time and memory limits. Even interpreted Python executes the computation instantly.

Test Cases

# helper: run solution on input string, return output string
import sys
import io

def solve():
    import sys
    input = sys.stdin.readline

    n = int(input())
    print((3 * n * n - n) // 2)

def run(inp: str) -> str:
    backup_stdin = sys.stdin
    backup_stdout = sys.stdout

    sys.stdin = io.StringIO(inp)
    sys.stdout = io.StringIO()

    solve()

    output = sys.stdout.getvalue()

    sys.stdin = backup_stdin
    sys.stdout = backup_stdout

    return output

# provided sample
assert run("2\n") == "5\n", "sample 1"

# custom cases
assert run("1\n") == "1\n", "minimum input"
assert run("3\n") == "12\n", "small odd value"
assert run("10\n") == "145\n", "larger value"
assert run("100\n") == "14950\n", "maximum constraint"

Test input	Expected output	What it validates
1	1	Minimum valid input
3	12	Correct arithmetic for small values
10	145	Formula works for larger numbers
100	14950	Maximum constraint handled correctly

Edge Cases

The smallest possible input is:

1

The algorithm computes:

$$\frac{3 \cdot 1^2 - 1}{2} = \frac{2}{2} = 1$$

The program prints:

1

This confirms there is no off-by-one mistake in indexing the sequence.

Another easy place to fail is operator precedence. Consider:

2

The correct evaluation is:

$$\frac{3 \cdot 2^2 - 2}{2} = 5$$

The implementation explicitly uses parentheses:

(3 * n * n - n) // 2

so the subtraction happens before division, preventing incorrect results.

The maximum input is:

100

The algorithm computes:

$$\frac{3 \cdot 100^2 - 100}{2} = \frac{30000 - 100}{2} = 14950$$

Python integers easily handle this range, so there is no overflow risk.