In the ancient land of Num-Burr existed a secret society whose beliefs were a strange mix of mathematics, logic, and mysticism. Potential new members had to undergo the Trial of the Gods.
Six of the Elders met with the applicant and each Elder made five statements, of which four were true and one was false. The statements referred to the Divine Qualities, the Sacred Numbers, and the Number of the Gods. The applicant had to logically determine the Number of the Gods. The applicant was confined to a prison cell with water, light, and writing materials. After three days he was brought before the Elders and asked for the Number of the Gods. If he answered correctly he was admitted to the society; if not he was executed. Here is a record of the statements, with the Elders designated by letters, and all Numbers are positive integers:
(A1) No two Sacred Numbers are equal.
(A2) The sum of the Sacred Numbers is not a perfect cube.
(A3) The Number of the Gods is a perfect cube.
(A4) All Sacred Numbers have the same Digital Sum.
(A5) Each Sacred Number has at least 2 digits but less than 5 digits.
(B1) At least 2 of the Sacred Numbers have the Divine Quality of being Palindromes.
(B2) At least 2 of the Sacred Numbers have the Divine Quality of having their digits in Strictly Increasing order (no 2 digits equal).
(B3) At least 3 of the Sacred Numbers have the Divine Quality of having their digits in Strictly Decreasing order (no 2 digits equal).
(B4) The total number of digits of all of the Sacred Numbers is less than 22.
(B5) All of the Sacred Numbers have Digital Sums greater than 18.
(C1) At least 1 of the Sacred Numbers has the Divine Quality of being Square.
(C2) At least 3 of the Sacred Numbers have the Divine Quality of being Triangular.
(C3) At least 3 of the Sacred Numbers have the Divine Quality of being Prime.
(C4) At least 2 of the Sacred Numbers possess 3 of the 6 Divine Qualities.
(C5) None of the Sacred Numbers has 4 digits.
(D1) The number of Sacred Numbers is not Square.
(D2) The number of Sacred Numbers is not Triangular.
(D3) The number of Sacred Numbers is not a perfect cube.
(D4) The sum of the Sacred Numbers is a perfect cube.
(D5) At least one of the Sacred Numbers has 2 digits.
(E1) No two Sacred Numbers have the same Digital Product.
(E2) Exactly 1 Sacred Number has a Digital Product which is Prime.
(E3) Exactly 1 Sacred Number has a Digital Product which is Triangular.
(E4) Exactly 3 Sacred Numbers have Digital Products which are Square.
(E5) No Sacred Number has a Digital Product which is less than 8.
(F1) Each Sacred Number has at least 3 digits.
(F2) The Number of the Gods is not a perfect cube.
(F3) The sum of the Sacred Numbers and the Number of the Gods are not both perfect cubes.
(F4) The Number of the Gods possesses none of the 6 Divine Qualities.
(F5) The Number of the Gods is equal to the sum of the Digital Products of all of the Sacred Numbers.
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In the ancient land of Num-Burr existed a secret society whose beliefs were a strange mix of mathematics, logic, and mysticism. Potential new members had to undergo the Trial of the Gods.
Six of the Elders met with the applicant and each Elder made five statements, of which four were true and one was false. The statements referred to the Divine Qualities, the Sacred Numbers, and the Number of the Gods. The applicant had to logically determine the Number of the Gods. The applicant was confined to a prison cell with water, light, and writing materials. After three days he was brought before the Elders and asked for the Number of the Gods. If he answered correctly he was admitted to the society; if not he was executed. Here is a record of the statements, with the Elders designated by letters, and all Numbers are positive integers:
(A1) No two Sacred Numbers are equal.
(A2) The sum of the Sacred Numbers is not a perfect cube.
(A3) The Number of the Gods is a perfect cube.
(A4) All Sacred Numbers have the same Digital Sum.
(A5) Each Sacred Number has at least 2 digits but less than 5 digits.
(B1) At least 2 of the Sacred Numbers have the Divine Quality of being Palindromes.
(B2) At least 2 of the Sacred Numbers have the Divine Quality of having their digits in Strictly Increasing order (no 2 digits equal).
(B3) At least 3 of the Sacred Numbers have the Divine Quality of having their digits in Strictly Decreasing order (no 2 digits equal).
(B4) The total number of digits of all of the Sacred Numbers is less than 22.
(B5) All of the Sacred Numbers have Digital Sums greater than 18.
(C1) At least 1 of the Sacred Numbers has the Divine Quality of being Square.
(C2) At least 3 of the Sacred Numbers have the Divine Quality of being Triangular.
(C3) At least 3 of the Sacred Numbers have the Divine Quality of being Prime.
(C4) At least 2 of the Sacred Numbers possess 3 of the 6 Divine Qualities.
(C5) None of the Sacred Numbers has 4 digits.
(D1) The number of Sacred Numbers is not Square.
(D2) The number of Sacred Numbers is not Triangular.
(D3) The number of Sacred Numbers is not a perfect cube.
(D4) The sum of the Sacred Numbers is a perfect cube.
(D5) At least one of the Sacred Numbers has 2 digits.
(E1) No two Sacred Numbers have the same Digital Product.
(E2) Exactly 1 Sacred Number has a Digital Product which is Prime.
(E3) Exactly 1 Sacred Number has a Digital Product which is Triangular.
(E4) Exactly 3 Sacred Numbers have Digital Products which are Square.
(E5) No Sacred Number has a Digital Product which is less than 8.
(F1) Each Sacred Number has at least 3 digits.
(F2) The Number of the Gods is not a perfect cube.
(F3) The sum of the Sacred Numbers and the Number of the Gods are not both perfect cubes.
(F4) The Number of the Gods possesses none of the 6 Divine Qualities.
(F5) The Number of the Gods is equal to the sum of the Digital Products of all of the Sacred Numbers.
What was the Number of the Gods?
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