KNOTT-6.2 — Cassini’s Identity (Fₙ₋₁Fₙ₊₁ − Fₙ² = (−1)ⁿ) & The 64 ↔ 65 / 63 Jigsaw Dissections¶
Curated Source & Catalog Provenance¶
Catalog ID |
|
Curated Source |
Dr. Ron Knott, Harder Fibonacci Puzzles (#jigsaw1) |
Source Attribution |
© 1996–2025 Dr. Ron Knott, University of Surrey — Explore Dr. Knott’s original exposition |
Exact SymPy Validator |
|
GEOMETOR Concordance |
Constructive Synthesis & Exact Proof¶
Giovanni Domenico Cassini’s 1680 determinant identity
\(F_{n-1} F_{n+1} - F_n^2 = (-1)^n\) turns into a dramatic visual paradox at
\(n = 6\) (\(5 \times 13 - 8^2 = +1\)), examined by Dr. Ron Knott
on fibpuzzles2.html (KnottPuzzles2) alongside Sam Loyd Jr. and
Henry Ernest Dudeney’s complementary 64 = 63 rearrangement
(Gardner1956, Horadam1962).
Left Assembly — 5 × 13 = 65 (+1 Gap): Rearranging the four pieces of an \(8 \times 8 = 64\) chessboard—triangles OB8P1 and P2QT5 (each area \(12\)) plus trapezoids B8R0QP1 and OP2T5T0 (each area \(20\))—along diagonal O \(\to\) Q separates slopes \(3/8 = 0.375\) (OP1) and \(2/5 = 0.400\) (OP2), opening thin +1 gap parallelogram OP1QP2 of exact signed area \(+1\).
Right Assembly — 63-Unit Notched Block (−1 Overlap): Sliding the same four pieces into two \(5 \times 6 = 30\) wings joined by a \(3 \times 1 = 3\) bridge (outer polygon area \(63\), pieces S0S1U1U0, U0U3S2, U3U2S4S3, and U1S5U2) crosses edges U0U1 and U0U3 over −1 overlap parallelogram U0U3U2U1, whose unsigned area is equal to \(1\) (\(64 - 1 = 63\)).
Machine-Verified Exact Symbolic Invariants¶
Invariant |
Exact Symbolic Value |
|---|---|
|
\(64\) |
|
\(65\) |
|
\(1\) |
|
\(63\) |
|
\(1\) |
|
\(1\) |
|
\(-1\) |