math-quiz/lib/engine/variable-parenthesis.ts

import type {
  DistributedVariableTerm,
  VariableParenthesisProblem,
  VariableParenthesisTerm,
  VariableTermChoice,
  VariableUserAnswer,
  VariableGradeResult,
} from "./variable-types";

const MIN_MULTIPLIER = -9;
const MAX_MULTIPLIER = -1;
const MIN_COEFFICIENT = 1;
const MAX_COEFFICIENT = 9;
const MAX_PROBLEM_COUNT = 9 * 9 * 9 * 2 * 2;

interface VariableProblemSeed {
  multiplier: number;
  operator: "+" | "-";
  firstTerm: VariableParenthesisTerm;
  secondTerm: VariableParenthesisTerm;
}

function randomInt(min: number, max: number) {
  return Math.floor(Math.random() * (max - min + 1)) + min;
}

function shuffle<T>(items: T[]) {
  const shuffled = [...items];

  for (let index = shuffled.length - 1; index > 0; index -= 1) {
    const targetIndex = randomInt(0, index);
    const current = shuffled[index];
    shuffled[index] = shuffled[targetIndex];
    shuffled[targetIndex] = current;
  }

  return shuffled;
}

function createConstantTerm(coefficient: number): VariableParenthesisTerm {
  return {
    kind: "constant",
    coefficient,
  };
}

function createVariableTerm(coefficient: number): VariableParenthesisTerm {
  return {
    kind: "variable",
    coefficient,
    variable: "x",
  };
}

function termKey(term: VariableParenthesisTerm) {
  return `${term.kind}:${term.coefficient}`;
}

export function formatVariableTermLatex(
  term: Pick<VariableParenthesisTerm, "kind" | "coefficient" | "variable">,
) {
  if (term.kind === "constant") {
    return `${term.coefficient}`;
  }

  if (term.coefficient === 1) {
    return "x";
  }

  if (term.coefficient === -1) {
    return "-x";
  }

  return `${term.coefficient}x`;
}

function formatParenthesisTermLatex(term: VariableParenthesisTerm) {
  return formatVariableTermLatex(term);
}

export function formatVariableExpressionLatex(
  terms: [DistributedVariableTerm, DistributedVariableTerm],
) {
  const [firstTerm, secondTerm] = terms;
  const operator = secondTerm.coefficient < 0 ? "-" : "+";
  const positiveSecondTerm = {
    ...secondTerm,
    coefficient: Math.abs(secondTerm.coefficient),
  };

  return `${firstTerm.latex} ${operator} ${formatVariableTermLatex(
    positiveSecondTerm,
  )}`;
}

function distributeTerm(
  multiplier: number,
  term: VariableParenthesisTerm,
  sign: 1 | -1,
): DistributedVariableTerm {
  const coefficient = multiplier * sign * term.coefficient;

  return {
    kind: term.kind,
    coefficient,
    variable: term.variable,
    latex: formatVariableTermLatex({ ...term, coefficient }),
  };
}

function createChoice(
  key: string,
  termIndex: 0 | 1,
  term: DistributedVariableTerm,
  coefficient: number,
): VariableTermChoice {
  const signKey = coefficient < 0 ? "negative" : "positive";

  return {
    id: `${key}:term-${termIndex}:${signKey}`,
    coefficient,
    variable: term.variable,
    latex: formatVariableTermLatex({ ...term, coefficient }),
  };
}

function createChoices(
  key: string,
  termIndex: 0 | 1,
  term: DistributedVariableTerm,
) {
  const absoluteCoefficient = Math.abs(term.coefficient);
  const choices: VariableTermChoice[] = [
    createChoice(key, termIndex, term, -absoluteCoefficient),
    createChoice(key, termIndex, term, absoluteCoefficient),
  ];

  return shuffle(choices);
}

function createProblem(seed: VariableProblemSeed): VariableParenthesisProblem {
  const { multiplier, operator, firstTerm, secondTerm } = seed;
  const key = [
    multiplier,
    termKey(firstTerm),
    operator,
    termKey(secondTerm),
  ].join(":");
  const correctTerms: [DistributedVariableTerm, DistributedVariableTerm] = [
    distributeTerm(multiplier, firstTerm, 1),
    distributeTerm(multiplier, secondTerm, operator === "-" ? -1 : 1),
  ];
  const choices: [VariableTermChoice[], VariableTermChoice[]] = [
    createChoices(key, 0, correctTerms[0]),
    createChoices(key, 1, correctTerms[1]),
  ];

  return {
    id: `variable-parenthesis:${key}`,
    key,
    multiplier,
    operator,
    firstTerm,
    secondTerm,
    correctTerms,
    choices,
    latexBefore: `${multiplier}\\left(${formatParenthesisTermLatex(
      firstTerm,
    )} ${operator} ${formatParenthesisTermLatex(secondTerm)}\\right)`,
    latexAfter: formatVariableExpressionLatex(correctTerms),
  };
}

function createAllProblemSeeds() {
  const seeds: VariableProblemSeed[] = [];

  for (
    let multiplier = MIN_MULTIPLIER;
    multiplier <= MAX_MULTIPLIER;
    multiplier += 1
  ) {
    for (
      let constant = MIN_COEFFICIENT;
      constant <= MAX_COEFFICIENT;
      constant += 1
    ) {
      for (
        let variableCoefficient = MIN_COEFFICIENT;
        variableCoefficient <= MAX_COEFFICIENT;
        variableCoefficient += 1
      ) {
        for (const operator of ["+", "-"] as const) {
          seeds.push({
            multiplier,
            operator,
            firstTerm: createVariableTerm(variableCoefficient),
            secondTerm: createConstantTerm(constant),
          });
          seeds.push({
            multiplier,
            operator,
            firstTerm: createConstantTerm(constant),
            secondTerm: createVariableTerm(variableCoefficient),
          });
        }
      }
    }
  }

  return seeds;
}

const ALL_PROBLEM_SEEDS = createAllProblemSeeds();

export function generateVariableParenthesisProblem(): VariableParenthesisProblem {
  const constant = randomInt(MIN_COEFFICIENT, MAX_COEFFICIENT);
  const variableCoefficient = randomInt(MIN_COEFFICIENT, MAX_COEFFICIENT);
  const variableFirst = Math.random() < 0.5;
  const operator = Math.random() < 0.5 ? "+" : "-";

  return createProblem({
    multiplier: randomInt(MIN_MULTIPLIER, MAX_MULTIPLIER),
    operator,
    firstTerm: variableFirst
      ? createVariableTerm(variableCoefficient)
      : createConstantTerm(constant),
    secondTerm: variableFirst
      ? createConstantTerm(constant)
      : createVariableTerm(variableCoefficient),
  });
}

export function generateVariableParenthesisSession(
  count = 10,
): VariableParenthesisProblem[] {
  if (!Number.isInteger(count) || count < 1) {
    throw new Error("count must be a positive integer.");
  }

  if (count > MAX_PROBLEM_COUNT) {
    throw new Error(`count cannot exceed ${MAX_PROBLEM_COUNT}.`);
  }

  return shuffle(ALL_PROBLEM_SEEDS).slice(0, count).map(createProblem);
}

function findSelectedChoice(
  problem: VariableParenthesisProblem,
  termIndex: 0 | 1,
  choiceId: string | null,
): DistributedVariableTerm | null {
  if (!choiceId) return null;

  const choice = problem.choices[termIndex].find((item) => item.id === choiceId);

  if (!choice) return null;

  return {
    kind: problem.correctTerms[termIndex].kind,
    coefficient: choice.coefficient,
    variable: choice.variable,
    latex: choice.latex,
  };
}

export function gradeVariableParenthesisAnswer(
  problem: VariableParenthesisProblem,
  answer: VariableUserAnswer,
): VariableGradeResult {
  const userAnswer: [
    DistributedVariableTerm | null,
    DistributedVariableTerm | null,
  ] = [
    findSelectedChoice(problem, 0, answer.selectedChoiceIds[0]),
    findSelectedChoice(problem, 1, answer.selectedChoiceIds[1]),
  ];
  const correct =
    userAnswer[0]?.coefficient === problem.correctTerms[0].coefficient &&
    userAnswer[1]?.coefficient === problem.correctTerms[1].coefficient;

  return {
    correct,
    userAnswer,
    correctAnswer: problem.correctTerms,
  };
}