File: smallest-index-with-equal-value/solution.py

Date: 2026-06-06

Time: 19:09

Purpose

This file solves LeetCode 2057 — Smallest Index With Equal Value. It owns the single responsibility of finding the smallest index i in a list where i % 10 == nums[i].

Key Components

smallest_index(nums: list[int]) -> int

The sole public function. Contract:

• Input: A list of integers, each in range [0, 9], representing a 0-indexed array.
• Output: The smallest index i such that i % 10 == nums[i], or -1 if no such index exists.
• Guarantee: Returns the *first* match due to left-to-right iteration with early return.

Patterns

Linear scan with early exit — the simplest possible approach for a "find first match" problem. No preprocessing, no data structures. enumerate gives both index and value in a single pass, and the function returns immediately on the first hit. This is idiomatic Python for problems where you need the first element satisfying a predicate.

The % 10 operation extracts the ones digit of the index, which is the only digit that matters since nums[i] is constrained to [0, 9]. For indices 0–9, i % 10 == i. For indices 10+, only the last digit of the index is compared.

Dependencies

Imports: None — pure standard library Python.

Imported by: The "Imported By" list in the prompt is misleading — those are test files for *other* problems that happen to share a common test harness import pattern. The actual consumer is smallest-index-with-equal-value/test_solution.py.

Flow

1. Iterate over nums with enumerate, yielding (i, val) pairs starting from index 0.

2. For each pair, check if i % 10 == val.

3. On the first match, return i immediately.

4. If the loop exhausts without a match, return -1.

Time complexity: O(n) worst case, O(1) best case (match at index 0). Space complexity: O(1).

Invariants

• The function always returns a deterministic result for a given input — no randomness, no state.
• The return value is either a valid index into nums or exactly -1.
• If a non-negative value is returned, returnvalue % 10 == nums[returnvalue] holds.
• No index smaller than the returned value satisfies the condition (guaranteed by left-to-right scan with early return).

Error Handling

None. The function trusts its caller to provide a valid list[int]. An empty list produces -1 (the for loop simply doesn't execute). No exceptions are raised or caught.

Topics to Explore

• [file] smallest-index-with-equal-value/test_solution.py — See which edge cases the test suite covers (empty list, all matches, no matches, large indices)
• [file] smallest-index-with-equal-value/plan.md — The approach reasoning and alternative strategies considered before implementation
• [function] fixed-point/solution.py:fixed_point — A structurally similar problem (find i where nums[i] == i) that uses binary search on a sorted array instead of linear scan
• [general] modular-arithmetic-in-index-problems — How % 10 constrains the search space: only indices ending in digit d can match nums[i] == d, which means at most 1 in 10 indices can match any given value

Beliefs

• smallest-index-returns-first-match — smallest_index returns the leftmost index satisfying i % 10 == nums[i], guaranteed by sequential iteration with early return
• smallest-index-sentinel-is-negative-one — The function returns -1 (not None or an exception) when no valid index exists, matching LeetCode's expected contract
• smallest-index-is-pure — The function has no side effects, no mutable state, and no dependencies beyond its input argument
• smallest-index-linear-time — Worst-case time complexity is O(n) with a single pass; no sorting or auxiliary data structures are used