To solve the problem of finding the number of ways to place two non-attacking kings on an 8x8 chessboard, we follow these steps:

Step 1: Total Pairs of Kings

The total number of ways to place two identical kings is given by combinations of 64 squares taken 2 at a time:
[ \binom{64}{2} = \frac{64 \times 63}{2} = 2016 ]

Step 2: Count Attacking Pairs

Kings attack adjacent squares (horizontal, vertical, diagonal). We calculate these:

• Horizontal: 8 rows × 7 adjacent pairs = 56
• Vertical: 8 columns ×7 adjacent pairs =56
• Diagonals: 2 directions (top-left→bottom-right and top-right→bottom-left), each with 7×7=49 pairs → 49×2=98

Total attacking pairs:
[56 +56 +98 =210]

Step3: Non-Attacking Pairs

Subtract attacking pairs from total pairs:
[2016 -210 =1806]

Answer: (\boxed{1806})

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