Random Continued Fraction Convergent Quiz Generator

Description

Construct finite simple continued fractions with signed integral first coefficients and positive tails, retaining exact rational convergents and one hidden prompt.

A generated practice exercise contains a prompt and an answer key under a specific rule set. A planted answer establishes one possible solution; only an explicit enumeration or uniqueness check establishes whether alternatives exist.

Construct finite simple continued fractions with signed integral first coefficients and positive tails, retaining exact rational convergents and one hidden prompt.

When to use this tool

Use this generator to create reproducible continued fraction convergent quiz cases from the supplied model or bank. Inspect the returned evidence before using cases in a test or exercise.

How to use the tool and how it works

Supply minimumFirst, maximumFirst, maximumTail, length in the field shapes below. Set seed when you need an explicit replay.
Run the tool and correct any input error. For generated controls, distinguish a successful fixture containing an invalid case from failure to generate the fixture.
Read coefficients, convergents, hiddenIndex, prompt. Use the conventions below. Keep answer-bearing fields separate when presenting an exercise.

Draw a signed first continued-fraction coefficient and positive tail coefficients, delegate convergents, then hide one convergent as null while retaining the complete answer sequence and coefficient bank.^[1]

Delegated operations used for this result: convergents from continued fraction. Their calculation or parsing conventions apply to the corresponding result fields.^[2]

Input field	Meaning and units	Accepted input
minimumFirst	Minimum First	Required. Safe integer.
maximumFirst	Maximum First	Required. Safe integer.
maximumTail	Maximum Tail	Required. Safe integer; minimum 1; maximum 32.
length	Length	Required. Safe integer; minimum 1; maximum 16.
seed	Optional numeric seed; takes precedence over context randomness. Without either, use the fixed tool.io default seed.	Optional. Safe integer.

Output field	Interpretation	Returned shape
coefficients	Coefficients	Required. Array of Safe integer.; minimum items 1; maximum items 16.
convergents	Convergents	Required. Array of Object; no unknown keys.; minimum items 1; maximum items 16.
convergents[].numerator	Numerator	Required. Text.
convergents[].denominator	Denominator	Required. Text.
hiddenIndex	Hidden Index	Required. Safe integer.
prompt	Prompt	Required. Array of Object; no unknown keys. or Null.; minimum items 1; maximum items 16.
randomness	Replay source and normalized 32-bit seed, or null seed for an advancing context source.	Required. Object; no unknown keys.
randomness.seed	Optional safe-integer replay seed; normalization and precedence are explained below.	Required. Safe integer. or Null.
randomness.source	Source	Required. Text; one of "context", "seed".

An explicit safe-integer seed takes precedence over injected context randomness and is normalized to an unsigned 32-bit integer. With no seed and no context source, replay uses the fixed tool.io default seed. A context source must return finite values from zero inclusive to one exclusive; its advancing state must be retained separately. randomness records the source and normalized seed, or null for context. These generators are not cryptographic random sources.

json

{
  "input": {
    "minimumFirst": 1,
    "maximumFirst": 3,
    "maximumTail": 4,
    "length": 3,
    "seed": 42
  },
  "output": {
    "coefficients": [
      2,
      2,
      4
    ],
    "convergents": [
      {
        "numerator": "2",
        "denominator": "1"
      },
      {
        "numerator": "5",
        "denominator": "2"
      },
      {
        "numerator": "22",
        "denominator": "9"
      }
    ],
    "hiddenIndex": 2,
    "prompt": [
      {
        "numerator": "2",
        "denominator": "1"
      },
      {
        "numerator": "5",
        "denominator": "2"
      },
      null
    ],
    "randomness": {
      "seed": 42,
      "source": "seed"
    }
  }
}

Limitations and assumptions

Coefficient order defines the continued fraction and its convergents. Bounds inherit the linked calculator's exact-integer contract; the exercise is generated from a known expansion rather than inferring one from arbitrary measurements.
A replay requires the same complete input and implementation. The stated sampling convention applies within its eligible space. A finite draw does not exhaust all possible outputs; repeats remain possible unless a distinct-selection rule is explicit.
The field tables state structural bounds; cross-field constraints and bounded search or output budgets also apply. Unknown keys are rejected where the object contract disallows them. Input errors are distinct from null fields and deliberate invalid controls in a successful output.

Alternative or Complementary methods

For a supplied example, inspect the returned construction or delegated check directly. To evaluate a consumer, also test cases chosen for its own boundaries; a generated finite corpus does not establish exhaustive coverage.

References

Tool behavior and example: Random Continued Fraction Convergent Quiz Generator — tool.io
Calculation or validation: math/convergents-from-continued-fraction — tool.io

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