###########################
#  Initialization

DOCUMENT();        
loadMacros(
  "PGstandard.pl",
  "MathObjects.pl",
  "PGML.pl",
  "PGbasicmacros.pl",
  "parserRadioButtons.pl",
  "draggableProof.pl",
  "PGcourse.pl"
);

TEXT(beginproblem());

###########################
#  Setup
Context("Numeric");
$CorrectProof = DraggableProof([   
"Let \(n\) be even.",
"By the definition of even, \(n = 2k\) for some integer \(k\).",
"It follows that  
\(\\n^2 + 3n + 5 = (2k)^2 + 3(2k) + 5\)
\(\\ \hspace{38mm}= 4k^2 + 6k + 5\)
\(\\ \hspace{37mm} = 2(2k^2 + 3k+ 2) + 1\)",
"Since \(k\) is an integer,",
"\(2k^2 + 3k+ 2\) is an integer.",
"Since \(n^2 + 3n + 5\) equals \(2\) times an integer plus one,",
"\(n^2 + 3n + 5\) is odd."],

#  The lines below are extras and will be listed as options but not
#  needed for the correct answer in $Proof.

["\(2k^2 + 3k+ 2\) is even.",
"By the definition of odd, \(n = 2k + 1\) for some integer \(k\).",
"Let \(n\) be odd."] ,

SourceLabel => "Choose from this list of sentences",
TargetLabel => "Direct proof of the statement (in order):",
);

###################################
# Main text

BEGIN_PGML

Order *[$CorrectProof->numNeeded] of* the following sentences so that they form a direct proof of the statement: If [`n`] is even, then [`n^2 + 3n + 5`] is odd.
[@ $CorrectProof->Print @]*

END_PGML

############################
#  Answer evaluation

$showPartialCorrectAnswers = 0;

install_problem_grader(~~&std_problem_grader);

ANS($CorrectProof->cmp);

ENDDOCUMENT();