This puzzle is borrowed from   math-frolic.blogspot.



Given the following twelve statements, which of them are true?

 1.  This is a numbered list of twelve statements.
 2.  Exactly 3 of the last 6 statements are true.
 3.  Exactly 2 of the even-numbered statements are true.
 4.  If statement 5 is true, then statements 6 and 7 are both true.
 5.  The 3 preceding statements are all false.
 6.  Exactly 4 of the odd-numbered statements are true.
 7.  Either statement 2 or 3 is true, but not both.
 8.  If statement 7 is true, then 5 and 6 are both true.
 9.  Exactly 3 of the first 6 statements are true.
10.  The next two statements are both true.
11.  Exactly 1 of statements 7, 8 and 9 are true.
12.  Exactly 4 of the preceding statements are true.



Task

When you get tired of trying to figure it out in your head,
write a program to solve it, and print the correct answer or answers.



Extra credit

Print out a table of near misses, that is, solutions that are contradicted by only a single statement.




% {{trans|Prolog}}
go ?=>
  puzzle,  
  fail, % check for more answers
  nl.
go => true.
 
puzzle =>
       % 1. This is a numbered list of twelve statements.
	L = [A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12],
	L :: 0..1,
	element(1, L, 1),
 
        % 2.  Exactly 3 of the last 6 statements are true.
	A2 #<=>  sum(L[7..12]) #= 3,        
 
        % 3.  Exactly 2 of the even-numbered statements are true.
	A3 #<=> sum([L[I]:I in 1..12,I mod 2 == 0]) #= 2,        
 
        % 4.  If statement 5 is true, then statements 6 and 7 are both true.
	A4 #<=> (A5 #=> (A6 #/\ A7)),
 
        % 5.  The 3 preceding statements are all false.
	A5 #<=> sum(L[2..4]) #= 0,        
 
        % 6.  Exactly 4 of the odd-numbered statements are true.
	A6 #<=> sum([L[I]:I in 1..12,I mod 2 == 1]) #= 4,
 
        % 7.  Either statement 2 or 3 is true, but not both.
	A7 #<=> (A2 + A3 #= 1),
 
        % 8.  If statement 7 is true, then 5 and 6 are both true.
	A8 #<=> (A7 #=>  A5 #/\ A6),
 
        % 9.  Exactly 3 of the first 6 statements are true.
	A9 #<=> sum(L[1..6]) #= 3,        
 
        % 10.  The next two statements are both true.
	A10 #<=> (A11 #/\ A12),
 
        % 11.  Exactly 1 of statements 7, 8 and 9 are true.
	A11 #<=> (A7 + A8 + A9 #= 1),
 
        % 12.  Exactly 4 of the preceding statements are true.
        A12 #<=> sum(L[1..11]) #= 4,
 
	solve(L),
 
        println('L'=L),
	printf("Statements %w are true.\n", [I.to_string : I in 1..12, L[I] == 1].join(" ")),
	nl.