Random Balance Scale Equation Puzzle Generator

Description

Construct balanced symbolic pan equations from sampled nonnegative weights, retaining exact solutions without claiming uniqueness.

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 balanced symbolic pan equations from sampled nonnegative weights, retaining exact solutions without claiming uniqueness.

When to use this tool

Use this generator to create reproducible balance scale equation puzzle 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 symbols, maximumWeight, equationCount 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 solution, equations. Use the conventions below. Keep answer-bearing fields separate when presenting an exercise.

Draw a nonnegative integer weight per symbol, then draw positive coefficients for every equation and calculate totals from that planted solution. Return all coefficient rows, totals, and the full symbol/weight answer.^[1]

Input field	Meaning and units	Accepted input
symbols	Symbols	Required. Array of Text; minimum UTF-16 length 1; maximum UTF-16 length 64.; minimum items 1; maximum items 8.
maximumWeight	Maximum Weight	Required. Safe integer; minimum 0; maximum 1000000.
equationCount	Equation Count	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
solution	Solution	Required. Array of Object; no unknown keys.; minimum items 1; maximum items 8.
solution[].symbol	Symbol	Required. Text; minimum UTF-16 length 1; maximum UTF-16 length 64.
solution[].weight	Weight	Required. Safe integer.
equations	Equations	Required. Array of Object; no unknown keys.; minimum items 1; maximum items 16.
equations[].coefficients	Coefficients	Required. Array of Safe integer.; minimum items 1; maximum items 8.
equations[].total	Total	Required. Safe integer.
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": {
    "symbols": [
      "x",
      "y"
    ],
    "maximumWeight": 5,
    "equationCount": 1,
    "seed": 42
  },
  "output": {
    "solution": [
      {
        "symbol": "x",
        "weight": 3
      },
      {
        "symbol": "y",
        "weight": 2
      }
    ],
    "equations": [
      {
        "coefficients": [
          3,
          3
        ],
        "total": 15
      }
    ],
    "randomness": {
      "seed": 42,
      "source": "seed"
    }
  }
}

Limitations and assumptions

A planted solution need not be unique or identifiable from the generated equations. This does not solve or rank-check the system, model a real balance scale, or assess exercise difficulty.
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 Balance Scale Equation Puzzle Generator — tool.io

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