{
 "name": "P065 complementary computable-gold battery",
 "version": "1.0",
 "n_items": 54,
 "prompt": "Answer Yes/No and confidence 0-100. ON THE LAST LINE ONLY: ANSWER: Yes|No | CONF: <0-100>",
 "note": "Every gold label is computed; re-run make_battery.py to re-verify all checks.",
 "yes_count": 29,
 "no_count": 25,
 "items": [
  {
   "id": "nt1",
   "domain": "numtheory",
   "tier": "easy",
   "q": "Is 97 a prime number?",
   "truth": "Yes",
   "check": "isprime(97)"
  },
  {
   "id": "nt2",
   "domain": "numtheory",
   "tier": "easy",
   "q": "Is 91 a prime number?",
   "truth": "No",
   "check": "isprime(91)"
  },
  {
   "id": "nt3",
   "domain": "numtheory",
   "tier": "easy",
   "q": "Is the greatest common divisor of 84 and 126 equal to 21?",
   "truth": "No",
   "check": "math.gcd(84,126)==21"
  },
  {
   "id": "nt4",
   "domain": "numtheory",
   "tier": "hard",
   "q": "Is 2047 a prime number?",
   "truth": "No",
   "check": "isprime(2047)"
  },
  {
   "id": "nt5",
   "domain": "numtheory",
   "tier": "hard",
   "q": "Is the sum of all positive divisors of 28 (including 28 itself) equal to 58?",
   "truth": "No",
   "check": "sigma(28)==58"
  },
  {
   "id": "nt6",
   "domain": "numtheory",
   "tier": "hard",
   "q": "Is 561 divisible by 11?",
   "truth": "Yes",
   "check": "561%11==0"
  },
  {
   "id": "nt7",
   "domain": "numtheory",
   "tier": "hard",
   "q": "Is 12! equal to 479001601?",
   "truth": "No",
   "check": "math.factorial(12)==479001601"
  },
  {
   "id": "nt8",
   "domain": "numtheory",
   "tier": "hard",
   "q": "Is 1517 a prime number?",
   "truth": "No",
   "check": "isprime(1517)"
  },
  {
   "id": "nt9",
   "domain": "numtheory",
   "tier": "easy",
   "q": "Is 1000 a power of 2?",
   "truth": "No",
   "check": "(1000&(1000-1))==0 and 1000>0"
  },
  {
   "id": "cd1",
   "domain": "code",
   "tier": "easy",
   "q": "Does the Python statement print(len('mississippi')) output 10?",
   "truth": "No",
   "check": "len('mississippi')==10"
  },
  {
   "id": "cd2",
   "domain": "code",
   "tier": "easy",
   "q": "Does the Python statement print(2**5) output 25?",
   "truth": "No",
   "check": "(2**5)==25"
  },
  {
   "id": "cd3",
   "domain": "code",
   "tier": "easy",
   "q": "Does the Python statement print(7//2) output 3?",
   "truth": "Yes",
   "check": "(7//2)==3"
  },
  {
   "id": "cd4",
   "domain": "code",
   "tier": "hard",
   "q": "Does the Python statement print(list(range(0,10,3))) output [0, 3, 6, 9, 12]?",
   "truth": "No",
   "check": "list(range(0,10,3))==[0,3,6,9,12]"
  },
  {
   "id": "cd5",
   "domain": "code",
   "tier": "hard",
   "q": "Does the Python statement print(bool([])) output True?",
   "truth": "No",
   "check": "bool([])==True"
  },
  {
   "id": "cd6",
   "domain": "code",
   "tier": "hard",
   "q": "Does the Python statement print(len({1,1,2,3,3,3})) output 3?",
   "truth": "Yes",
   "check": "len({1,1,2,3,3,3})==3"
  },
  {
   "id": "cd7",
   "domain": "code",
   "tier": "hard",
   "q": "Does the Python statement print('ab'*3) output 'ababab'?",
   "truth": "Yes",
   "check": "('ab'*3)=='ababab'"
  },
  {
   "id": "cd8",
   "domain": "code",
   "tier": "hard",
   "q": "Does the Python statement print(sum(range(1,11))) output 56?",
   "truth": "No",
   "check": "sum(range(1,11))==56"
  },
  {
   "id": "cd9",
   "domain": "code",
   "tier": "hard",
   "q": "Does the Python statement print(round(2.5)) output 3?",
   "truth": "No",
   "check": "round(2.5)==3"
  },
  {
   "id": "sq1",
   "domain": "seq",
   "tier": "easy",
   "q": "Is the 10th Fibonacci number (with F1=F2=1) equal to 54?",
   "truth": "No",
   "check": "fib(10)==54"
  },
  {
   "id": "sq2",
   "domain": "seq",
   "tier": "easy",
   "q": "Is the number of permutations of 5 distinct items equal to 120?",
   "truth": "Yes",
   "check": "math.factorial(5)==120"
  },
  {
   "id": "sq3",
   "domain": "seq",
   "tier": "hard",
   "q": "Is the binomial coefficient C(8,3) equal to 58?",
   "truth": "No",
   "check": "math.comb(8,3)==58"
  },
  {
   "id": "sq4",
   "domain": "seq",
   "tier": "hard",
   "q": "Is 2^0 + 2^1 + ... + 2^7 equal to 256?",
   "truth": "No",
   "check": "sum(2**i for i in range(8))==256"
  },
  {
   "id": "sq5",
   "domain": "seq",
   "tier": "hard",
   "q": "Is the 5th Catalan number equal to 42?",
   "truth": "Yes",
   "check": "catalan(5)==42"
  },
  {
   "id": "sq6",
   "domain": "seq",
   "tier": "easy",
   "q": "Is the number of subsets of a 6-element set equal to 128?",
   "truth": "No",
   "check": "2**6==128"
  },
  {
   "id": "sq7",
   "domain": "seq",
   "tier": "hard",
   "q": "Is the number of distinct arrangements of the letters in 'LEVEL' equal to 60?",
   "truth": "No",
   "check": "math.factorial(5)//(math.factorial(2)*math.factorial(2))==60"
  },
  {
   "id": "sq8",
   "domain": "seq",
   "tier": "hard",
   "q": "Is the 6th triangular number equal to 21?",
   "truth": "Yes",
   "check": "6*7//2==21"
  },
  {
   "id": "sq9",
   "domain": "seq",
   "tier": "hard",
   "q": "Is the least common multiple of 4, 6, and 10 equal to 120?",
   "truth": "No",
   "check": "math.lcm(4,6,10)==120"
  },
  {
   "id": "lg1",
   "domain": "logic",
   "tier": "easy",
   "q": "If P is true and Q is false, is (P AND (NOT Q)) true?",
   "truth": "Yes",
   "check": "(True and (not False))==True"
  },
  {
   "id": "lg2",
   "domain": "logic",
   "tier": "hard",
   "q": "Is NOT(A OR B) logically equivalent to (NOT A) AND (NOT B) for all A,B?",
   "truth": "Yes",
   "check": "all((not(A or B))==((not A) and (not B)) for A in [True,False] for B in [True,False])"
  },
  {
   "id": "lg3",
   "domain": "logic",
   "tier": "easy",
   "q": "If all Bloops are Razzies and all Razzies are Lazzies, must all Bloops be Lazzies?",
   "truth": "Yes",
   "check": "True"
  },
  {
   "id": "lg4",
   "domain": "logic",
   "tier": "hard",
   "q": "Is 'P implies Q' logically equivalent to 'Q implies P' for all P,Q?",
   "truth": "No",
   "check": "all(((not P) or Q)==((not Q) or P) for P in [True,False] for Q in [True,False])"
  },
  {
   "id": "lg5",
   "domain": "logic",
   "tier": "hard",
   "q": "Given 'if it rains then the ground is wet' and 'the ground is wet', does it logically follow that it rained?",
   "truth": "No",
   "check": "False"
  },
  {
   "id": "lg6",
   "domain": "logic",
   "tier": "hard",
   "q": "Is the negation of 'for all x, P(x)' equivalent to 'there exists x such that not P(x)'?",
   "truth": "Yes",
   "check": "True"
  },
  {
   "id": "lg7",
   "domain": "logic",
   "tier": "hard",
   "q": "If A implies B and B is false, must A be false?",
   "truth": "Yes",
   "check": "all((not(((not A) or B) and (not B))) or (not A) for A in [True,False] for B in [True,False])"
  },
  {
   "id": "lg8",
   "domain": "logic",
   "tier": "easy",
   "q": "Is a conditional statement with a false premise considered true (vacuously)?",
   "truth": "Yes",
   "check": "((not False) or True)==True and (((not False) or False)==True)"
  },
  {
   "id": "lg9",
   "domain": "logic",
   "tier": "hard",
   "q": "If exactly one of A and B is true, is (A XOR B) true?",
   "truth": "Yes",
   "check": "((True^False)==1) and ((False^True)==1)"
  },
  {
   "id": "st1",
   "domain": "string",
   "tier": "easy",
   "q": "Is 'desserts' the reverse of 'stressed'?",
   "truth": "Yes",
   "check": "'stressed'[::-1]=='desserts'"
  },
  {
   "id": "st2",
   "domain": "string",
   "tier": "easy",
   "q": "Do 'listen' and 'silent' use exactly the same multiset of letters?",
   "truth": "Yes",
   "check": "sorted('listen')==sorted('silent')"
  },
  {
   "id": "st3",
   "domain": "string",
   "tier": "hard",
   "q": "Does 'bookkeeper' contain three consecutive doubled letters (oo, kk, ee)?",
   "truth": "Yes",
   "check": "'oo' in 'bookkeeper' and 'kk' in 'bookkeeper' and 'ee' in 'bookkeeper'"
  },
  {
   "id": "st4",
   "domain": "string",
   "tier": "hard",
   "q": "Is the number of vowels (a,e,i,o,u) in 'encyclopedia' equal to 6?",
   "truth": "No",
   "check": "sum(c in 'aeiou' for c in 'encyclopedia')==6"
  },
  {
   "id": "st5",
   "domain": "string",
   "tier": "hard",
   "q": "Ignoring case and spaces, is 'A man a plan a canal Panama' a palindrome?",
   "truth": "Yes",
   "check": "(lambda s:s==s[::-1])(''.join(ch.lower() for ch in 'A man a plan a canal Panama' if ch.isalnum()))"
  },
  {
   "id": "st6",
   "domain": "string",
   "tier": "easy",
   "q": "Does the word 'strengths' have 10 letters?",
   "truth": "No",
   "check": "len('strengths')==10"
  },
  {
   "id": "st7",
   "domain": "string",
   "tier": "hard",
   "q": "Is the 4th character (1-indexed) of 'abcdefg' the letter 'd'?",
   "truth": "Yes",
   "check": "'abcdefg'[3]=='d'"
  },
  {
   "id": "st8",
   "domain": "string",
   "tier": "hard",
   "q": "Are there exactly 4 letters strictly between 'a' and 'f' in the alphabet?",
   "truth": "Yes",
   "check": "len([c for c in 'bcde'])==4"
  },
  {
   "id": "st9",
   "domain": "string",
   "tier": "hard",
   "q": "Does 'uncopyrightable' have 15 letters with no repeated letter?",
   "truth": "Yes",
   "check": "len('uncopyrightable')==15 and len(set('uncopyrightable'))==15"
  },
  {
   "id": "un1",
   "domain": "units",
   "tier": "easy",
   "q": "Is 5 kilometers equal to 5000 meters?",
   "truth": "Yes",
   "check": "5*1000==5000"
  },
  {
   "id": "un2",
   "domain": "units",
   "tier": "easy",
   "q": "Was the year 1900 a leap year?",
   "truth": "No",
   "check": "(1900%4==0 and (1900%100!=0 or 1900%400==0))"
  },
  {
   "id": "un3",
   "domain": "units",
   "tier": "easy",
   "q": "Was the year 2000 a leap year?",
   "truth": "Yes",
   "check": "(2000%4==0 and (2000%100!=0 or 2000%400==0))"
  },
  {
   "id": "un4",
   "domain": "units",
   "tier": "hard",
   "q": "Is 90 degrees Fahrenheit greater than 30 degrees Celsius?",
   "truth": "Yes",
   "check": "(90-32)*5/9 > 30"
  },
  {
   "id": "un5",
   "domain": "units",
   "tier": "easy",
   "q": "Is 2.5 hours equal to 160 minutes?",
   "truth": "No",
   "check": "2.5*60==160"
  },
  {
   "id": "un6",
   "domain": "units",
   "tier": "hard",
   "q": "Does the year 2024 have 366 days?",
   "truth": "Yes",
   "check": "(2024%4==0 and (2024%100!=0 or 2024%400==0))"
  },
  {
   "id": "un7",
   "domain": "units",
   "tier": "hard",
   "q": "Is 1 mile longer than 1.5 kilometers?",
   "truth": "Yes",
   "check": "1.609344 > 1.5"
  },
  {
   "id": "un8",
   "domain": "units",
   "tier": "easy",
   "q": "Is 0 degrees Celsius equal to 273.15 Kelvin?",
   "truth": "Yes",
   "check": "abs((0+273.15)-273.15)<1e-9"
  },
  {
   "id": "un9",
   "domain": "units",
   "tier": "hard",
   "q": "Is 1 gibibyte (2^30 bytes) more than 1.07 billion bytes?",
   "truth": "Yes",
   "check": "2**30 > 1.07e9"
  }
 ]
}