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Return a single integer only. The problem has been revised so that no final answer is an irrational number, a parameterized expression, a range, a proof, a drawing, a graph, a locus equation, or a region description.
A square number means the square of a nonnegative integer. Let
Find the number of positive integers for which is a square number.
We need n²+12n-12 = k².
n²+12n-12 = (n+6)²-48 = k².
So (n+6)²-k²=48. (n+6-k)(n+6+k)=48.
Let a=n+6-k, b=n+6+k, ab=48, a+b=2(n+6), same parity.
Both even: a·b=48 with both even. Factor pairs (positive, a≤b): (2,24): sum26, n+6=13, n=7, k=11. Check: 49+84-12=121=11². Yes. (4,12): sum16, n+6=8, n=2, k=4. Check:4+24-12=16=4². Yes. (6,8): sum14, n+6=7, n=1, k=1. Check:1+12-12=1. Yes.
Also negative factors but n positive so n+6 positive, a+b positive. k≥0.
Consider k could make a negative if k>n+6. Then a negative, b positive, product negative ≠48. So both positive.
Both odd: 48 not product of two odds (48 even). No.
So three values: n=1,2,7.
\boxed{3}
Comparative Analysis (P17) (Average across runs)
Averages use reported measurements only. Missing usage is not treated as zero.