File size: 4,726 Bytes
ab59633 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 | """Math and formal reasoning: one-shot problems with an exact answer and a single-field action."""
import itertools
import math
from fractions import Fraction
from pydantic import Field
from ..core import Action, ProxyTask, Step
class AnswerAction(Action):
answer: str = Field(description="Your final answer, exactly in the requested format. One reply only.")
def norm(text):
return str(text).strip().lower().replace(" ", "").rstrip(".")
class OneShotMath(ProxyTask):
family = "math-oneshot"
domain = "mathematics-or-formal-reasoning"
Action = AnswerAction
max_steps = 1
finish_action = {"answer": ""}
def setup(self):
r = self.rng
v = self.seed % 5
if v == 0: # lattice paths avoiding a blocked point
w, h = r.randint(4, 7), r.randint(4, 7)
bx, by = r.randint(1, w - 1), r.randint(1, h - 1)
total = math.comb(w + h, w) - math.comb(bx + by, bx) * math.comb(w - bx + h - by, w - bx)
self.text = (f"How many shortest grid paths go from (0,0) to ({w},{h}) moving only right or up, "
f"without passing through the point ({bx},{by})? Answer with an integer.")
self.truth = str(total)
elif v == 1: # probability as a reduced fraction
red, blue = r.randint(3, 7), r.randint(3, 7)
k = 3
p = Fraction(math.comb(red, 2) * blue, math.comb(red + blue, k))
self.text = (f"A bag has {red} red and {blue} blue marbles. Three are drawn without replacement. What "
f"is the probability that exactly two are red? Answer as a reduced fraction a/b.")
self.truth = f"{p.numerator}/{p.denominator}"
elif v == 2: # bounded integer solutions
n, k, cap = r.randint(10, 16), 3, r.randint(5, 7)
count = sum(1 for xs in itertools.product(range(cap + 1), repeat=k) if sum(xs) == n)
self.text = (f"How many ordered triples of integers (x, y, z) with 0 <= x, y, z <= {cap} satisfy "
f"x + y + z = {n}? Answer with an integer.")
self.truth = str(count)
elif v == 3: # last digits of a power
a, b = r.choice([3, 7, 13, 17]), r.randint(200, 999)
self.text = f"What are the last two digits of {a}^{b}? Answer with two digits, e.g. 07."
self.truth = f"{pow(a, b, 100):02d}"
else: # knights and knaves
while True:
truth = [r.random() < 0.5 for _ in range(3)]
names = ["A", "B", "C"]
stmts = []
for i in range(3):
j = r.choice([x for x in range(3) if x != i])
kind = r.choice(["is_knave", "same", "count"])
stmts.append((i, j, kind, r.randint(1, 2)))
def holds(world, s):
i, j, kind, c = s
if kind == "is_knave":
return not world[j]
if kind == "same":
return world[i] == world[j]
return sum(world) == c
worlds = [w for w in itertools.product([True, False], repeat=3)
if all(holds(w, s) == w[s[0]] for s in stmts)]
if len(worlds) == 1:
break
say = {"is_knave": lambda i, j, c: f"{names[j]} is a knave.",
"same": lambda i, j, c: f"{names[j]} and I are the same type.",
"count": lambda i, j, c: f"Exactly {c} of us {'is a knight' if c == 1 else 'are knights'}."}
lines = " ".join(f"{names[i]} says: \"{say[kind](i, j, c)}\"" for i, j, kind, c in stmts)
self.text = (f"On an island, knights always tell the truth and knaves always lie. {lines} Which of A, "
f"B, C are knights? Answer with the letters in alphabetical order separated by commas, "
f"or 'none'.")
knights = [names[i] for i in range(3) if worlds[0][i]]
self.truth = ",".join(knights) if knights else "none"
def reset(self):
return {"problem": self.text}
def act(self, a):
got = norm(a.answer)
ok = got == norm(self.truth)
if not ok and "/" in self.truth:
try:
ok = Fraction(got) == Fraction(self.truth) and "/" in got
ok = ok and Fraction(got).denominator == int(got.split("/")[1]) # must be reduced
except (ValueError, ZeroDivisionError):
ok = False
return Step({"problem": "", "result": "recorded"}, True, 1.0 if ok else 0.0)
def oracle(self):
return [{"answer": self.truth}]
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