diff --git a/examples/benchmarks/mathematical.py b/examples/benchmarks/mathematical.py
new file mode 100644
index 00000000..41c9163d
--- /dev/null
+++ b/examples/benchmarks/mathematical.py
@@ -0,0 +1,263 @@
+"""Benchmarks based on Vaguery's mathematical programming challenges.
+
+The scoring functions use :class:`fractions.Fraction` deliberately.  These
+problems are about exact rational arithmetic and floating-point fitness would
+make correct programs appear incorrect for sufficiently large inputs.
+"""
+
+from __future__ import annotations
+
+from abc import ABC
+from dataclasses import dataclass
+from fractions import Fraction
+from typing import Annotated
+
+from examples.benchmarks.benchmark import Benchmark
+from geneticengine.grammar.grammar import Grammar, extract_grammar
+from geneticengine.grammar.metahandlers.ints import IntRange
+from geneticengine.problems import Problem, SingleObjectiveProblem
+
+
+class IntegerExpression(ABC):
+    def evaluate(self, values: tuple[int, ...]) -> int:
+        raise NotImplementedError
+
+
+@dataclass
+class IntegerLiteral(IntegerExpression):
+    value: Annotated[int, IntRange(-100, 100)]
+
+    def evaluate(self, values: tuple[int, ...]) -> int:
+        return self.value
+
+
+@dataclass
+class InputInteger(IntegerExpression):
+    index: Annotated[int, IntRange(0, 1)]
+
+    def evaluate(self, values: tuple[int, ...]) -> int:
+        return values[self.index]
+
+
+@dataclass
+class Add(IntegerExpression):
+    left: IntegerExpression
+    right: IntegerExpression
+
+    def evaluate(self, values: tuple[int, ...]) -> int:
+        return self.left.evaluate(values) + self.right.evaluate(values)
+
+
+@dataclass
+class Subtract(IntegerExpression):
+    left: IntegerExpression
+    right: IntegerExpression
+
+    def evaluate(self, values: tuple[int, ...]) -> int:
+        return self.left.evaluate(values) - self.right.evaluate(values)
+
+
+@dataclass
+class Multiply(IntegerExpression):
+    left: IntegerExpression
+    right: IntegerExpression
+
+    def evaluate(self, values: tuple[int, ...]) -> int:
+        return self.left.evaluate(values) * self.right.evaluate(values)
+
+
+@dataclass
+class Quotient(IntegerExpression):
+    left: IntegerExpression
+    right: IntegerExpression
+
+    def evaluate(self, values: tuple[int, ...]) -> int:
+        denominator = self.right.evaluate(values)
+        return 0 if denominator == 0 else self.left.evaluate(values) // denominator
+
+
+@dataclass
+class Remainder(IntegerExpression):
+    left: IntegerExpression
+    right: IntegerExpression
+
+    def evaluate(self, values: tuple[int, ...]) -> int:
+        denominator = self.right.evaluate(values)
+        return 0 if denominator == 0 else self.left.evaluate(values) % denominator
+
+
+@dataclass
+class FiveUnitFractions:
+    terms: tuple[IntegerExpression, IntegerExpression, IntegerExpression, IntegerExpression, IntegerExpression]
+
+    def evaluate(self, values: tuple[int, ...]) -> tuple[int, ...]:
+        return tuple(term.evaluate(values) for term in self.terms)
+
+
+@dataclass
+class IntegerPair:
+    numerator: IntegerExpression
+    denominator: IntegerExpression
+
+    def evaluate(self, values: tuple[int, ...]) -> tuple[int, int]:
+        return self.numerator.evaluate(values), self.denominator.evaluate(values)
+
+
+def fixed_egyptian_fraction(k: int) -> tuple[int, ...]:
+    """Return five distinct nonzero denominators summing to ``1/k``."""
+    return 2 * k, 3 * k, 6 * k, 7 * k, -7 * k
+
+
+def rational_egyptian_fraction(numerator: int, denominator: int) -> tuple[int, ...]:
+    """Find five signed unit fractions for a small rational target.
+
+    The finite benchmark domain is intentionally small; exhaustive search is
+    used for the reference oracle so the benchmark remains independently
+    checkable and does not encode a particular synthesis strategy.
+    """
+    target = Fraction(numerator, denominator)
+    forbidden = {numerator, denominator}
+    candidates = [i for i in range(-200, 201) if i and i not in forbidden]
+    for a in candidates:
+        for b in candidates:
+            if b == a:
+                continue
+            remainder = target - Fraction(1, a) - Fraction(1, b)
+            if remainder.numerator not in (-1, 1):
+                continue
+            c = remainder.denominator * remainder.numerator
+            if c in forbidden or c in {a, b} or c not in candidates:
+                continue
+            d = next((x for x in candidates if x not in {a, b, c} and -x in candidates), None)
+            if d is not None and -d not in {a, b, c, d}:
+                return a, b, c, d, -d
+    raise ValueError(f"no five-term representation found for {numerator}/{denominator}")
+
+
+def reptend_fraction(prefix: int, repetend: int) -> tuple[int, int]:
+    prefix_digits = len(str(prefix))
+    repetend_digits = len(str(repetend))
+    cycle = 10**repetend_digits - 1
+    return prefix * cycle + repetend, 10**prefix_digits * cycle
+
+
+def incremented_reciprocal(n: int) -> tuple[int, int]:
+    """Increment every digit of the repeating decimal expansion of ``1/n``."""
+    if n <= 0:
+        raise ValueError("n must be positive")
+    remainder = 1 % n
+    seen: dict[int, int] = {}
+    digits: list[int] = []
+    while remainder and remainder not in seen:
+        seen[remainder] = len(digits)
+        remainder *= 10
+        digits.append(remainder // n)
+        remainder %= n
+    cycle_start = seen.get(remainder, len(digits))
+    nonrepeating = digits[:cycle_start]
+    repeating = digits[cycle_start:] or [0]
+    shifted_nonrepeating = [((digit + 1) % 10) for digit in nonrepeating]
+    shifted_repeating = [((digit + 1) % 10) for digit in repeating]
+    scale = 10 ** len(shifted_nonrepeating)
+    prefix = 0
+    for digit in shifted_nonrepeating:
+        prefix = 10 * prefix + digit
+    cycle_value = 0
+    for digit in shifted_repeating:
+        cycle_value = 10 * cycle_value + digit
+    denominator = scale * (10 ** len(shifted_repeating) - 1)
+    value = Fraction(prefix, scale) + Fraction(cycle_value, denominator)
+    return value.numerator, value.denominator
+
+
+def _unit_fraction_error(candidate: tuple[int, ...], target: Fraction, forbidden: set[int]) -> float:
+    if len(candidate) != 5 or any(value == 0 for value in candidate):
+        return 1_000_000.0
+    error = abs(sum((Fraction(1, value) for value in candidate), Fraction()) - target)
+    penalty = 0 if len(set(candidate)) == 5 and not forbidden.intersection(candidate) else 1
+    return float(error) + penalty
+
+
+def _pair_error(candidate: tuple[int, int], target: Fraction) -> float:
+    numerator, denominator = candidate
+    if denominator == 0:
+        return 1_000_000.0
+    return float(abs(Fraction(numerator, denominator) - target))
+
+
+class _IntegerExpressionBenchmark(Benchmark):
+    expression_nodes = [IntegerLiteral, InputInteger, Add, Subtract, Multiply, Quotient, Remainder]
+
+    def get_grammar(self) -> Grammar:
+        return extract_grammar(self.expression_nodes + [self.root_type], self.root_type)
+
+
+class FixedEgyptianFractionBenchmark(_IntegerExpressionBenchmark):
+    root_type = FiveUnitFractions
+
+    def __init__(self) -> None:
+        self.problem = SingleObjectiveProblem(
+            minimize=True,
+            target=0,
+            fitness_function=lambda candidate: _unit_fraction_error(
+                candidate.evaluate((7, 0)), Fraction(1, 7), {7}
+            ),
+        )
+
+    def get_problem(self) -> Problem:
+        return self.problem
+
+
+class RationalEgyptianFractionBenchmark(_IntegerExpressionBenchmark):
+    root_type = FiveUnitFractions
+
+    def __init__(self) -> None:
+        target = Fraction(5, 12)
+        self.problem = SingleObjectiveProblem(
+            minimize=True,
+            target=0,
+            fitness_function=lambda candidate: _unit_fraction_error(
+                candidate.evaluate((5, 12)), target, {5, 12}
+            ),
+        )
+
+    def get_problem(self) -> Problem:
+        return self.problem
+
+
+class ReptendFractionBenchmark(_IntegerExpressionBenchmark):
+    root_type = IntegerPair
+
+    def __init__(self) -> None:
+        target = Fraction(*reptend_fraction(539, 4762))
+        self.problem = SingleObjectiveProblem(
+            minimize=True,
+            target=0,
+            fitness_function=lambda candidate: _pair_error(candidate.evaluate((539, 4762)), target),
+        )
+
+    def get_problem(self) -> Problem:
+        return self.problem
+
+
+class IncrementedReciprocalBenchmark(_IntegerExpressionBenchmark):
+    root_type = IntegerPair
+
+    def __init__(self) -> None:
+        target = Fraction(*incremented_reciprocal(7))
+        self.problem = SingleObjectiveProblem(
+            minimize=True,
+            target=0,
+            fitness_function=lambda candidate: _pair_error(candidate.evaluate((7, 0)), target),
+        )
+
+    def get_problem(self) -> Problem:
+        return self.problem
+
+
+BENCHMARKS = [
+    FixedEgyptianFractionBenchmark,
+    RationalEgyptianFractionBenchmark,
+    ReptendFractionBenchmark,
+    IncrementedReciprocalBenchmark,
+]
diff --git a/examples/mathematical_programming/__init__.py b/examples/mathematical_programming/__init__.py
new file mode 100644
index 00000000..b2f2afe6
--- /dev/null
+++ b/examples/mathematical_programming/__init__.py
@@ -0,0 +1 @@
+"""Examples for the mathematical programming challenges."""
diff --git a/examples/mathematical_programming/fixed_egyptian_fractions.py b/examples/mathematical_programming/fixed_egyptian_fractions.py
new file mode 100644
index 00000000..f8f6b923
--- /dev/null
+++ b/examples/mathematical_programming/fixed_egyptian_fractions.py
@@ -0,0 +1,8 @@
+"""Genetic programming example for fixed-size Egyptian fractions."""
+
+from examples.benchmarks.benchmark import example_run
+from examples.benchmarks.mathematical import FixedEgyptianFractionBenchmark
+
+
+if __name__ == "__main__":
+    example_run(FixedEgyptianFractionBenchmark())
diff --git a/examples/mathematical_programming/incremented_reciprocal.py b/examples/mathematical_programming/incremented_reciprocal.py
new file mode 100644
index 00000000..9097ed4a
--- /dev/null
+++ b/examples/mathematical_programming/incremented_reciprocal.py
@@ -0,0 +1,8 @@
+"""Genetic programming example for the incremented reciprocal challenge."""
+
+from examples.benchmarks.benchmark import example_run
+from examples.benchmarks.mathematical import IncrementedReciprocalBenchmark
+
+
+if __name__ == "__main__":
+    example_run(IncrementedReciprocalBenchmark())
diff --git a/examples/mathematical_programming/rational_egyptian_fractions.py b/examples/mathematical_programming/rational_egyptian_fractions.py
new file mode 100644
index 00000000..b17d74e3
--- /dev/null
+++ b/examples/mathematical_programming/rational_egyptian_fractions.py
@@ -0,0 +1,8 @@
+"""Genetic programming example for rational-target Egyptian fractions."""
+
+from examples.benchmarks.benchmark import example_run
+from examples.benchmarks.mathematical import RationalEgyptianFractionBenchmark
+
+
+if __name__ == "__main__":
+    example_run(RationalEgyptianFractionBenchmark())
diff --git a/examples/mathematical_programming/reptend_fraction.py b/examples/mathematical_programming/reptend_fraction.py
new file mode 100644
index 00000000..9e5aff41
--- /dev/null
+++ b/examples/mathematical_programming/reptend_fraction.py
@@ -0,0 +1,8 @@
+"""Genetic programming example for repeating-decimal rational numbers."""
+
+from examples.benchmarks.benchmark import example_run
+from examples.benchmarks.mathematical import ReptendFractionBenchmark
+
+
+if __name__ == "__main__":
+    example_run(ReptendFractionBenchmark())
diff --git a/run_examples.sh b/run_examples.sh
index 9347c1e8..6bcaf72c 100755
--- a/run_examples.sh
+++ b/run_examples.sh
@@ -66,6 +66,11 @@ run_example examples/benchmarks/santafe.py
 run_example examples/benchmarks/string_match.py
 run_example examples/benchmarks/vectorialgp.py
 
+run_example examples/mathematical_programming/fixed_egyptian_fractions.py
+run_example examples/mathematical_programming/rational_egyptian_fractions.py
+run_example examples/mathematical_programming/reptend_fraction.py
+run_example examples/mathematical_programming/incremented_reciprocal.py
+
 
 run_example examples/progsys/Number_IO.py
 run_example examples/progsys/Median.py
diff --git a/tests/benchmarks/test_mathematical.py b/tests/benchmarks/test_mathematical.py
new file mode 100644
index 00000000..3db5afdd
--- /dev/null
+++ b/tests/benchmarks/test_mathematical.py
@@ -0,0 +1,46 @@
+from fractions import Fraction
+
+from examples.benchmarks.mathematical import FixedEgyptianFractionBenchmark
+from examples.benchmarks.mathematical import IncrementedReciprocalBenchmark
+from examples.benchmarks.mathematical import RationalEgyptianFractionBenchmark
+from examples.benchmarks.mathematical import ReptendFractionBenchmark
+from examples.benchmarks.mathematical import fixed_egyptian_fraction
+from examples.benchmarks.mathematical import incremented_reciprocal
+from examples.benchmarks.mathematical import reptend_fraction
+
+
+def test_fixed_egyptian_fraction_is_valid():
+    terms = fixed_egyptian_fraction(7)
+    assert len(set(terms)) == 5
+    assert Fraction(1, 7) == sum((Fraction(1, term) for term in terms), Fraction())
+
+
+def test_reptend_example():
+    numerator, denominator = reptend_fraction(539, 4762)
+    assert Fraction(numerator, denominator) == Fraction(539 * 9999 + 4762, 1000 * 9999)
+
+
+def test_incremented_reciprocal_examples():
+    assert incremented_reciprocal(1) == (1, 9)
+    assert incremented_reciprocal(2) == (11, 18)
+    # The page prints 16/163, but 0.overline{253968} reduces to 16/63.
+    assert incremented_reciprocal(7) == (16, 63)
+
+
+def test_benchmarks_expose_zero_fitness_reference_targets():
+    fixed = FixedEgyptianFractionBenchmark()
+    candidate = type("C", (), {"evaluate": staticmethod(lambda _: fixed_egyptian_fraction(7))})()
+    assert fixed.get_problem().evaluate(candidate).fitness_components == [0]
+    reptend = ReptendFractionBenchmark()
+    assert reptend.get_problem().target == [0]
+    assert incremented_reciprocal(7) == (16, 63)
+
+
+def test_all_benchmark_grammars_are_constructible():
+    for benchmark in (
+        FixedEgyptianFractionBenchmark(),
+        RationalEgyptianFractionBenchmark(),
+        ReptendFractionBenchmark(),
+        IncrementedReciprocalBenchmark(),
+    ):
+        assert benchmark.get_grammar() is not None