MoMath × 3Blue1Brown

Peg Jump Square Puzzle

Posed by @3blue1brown: Can pegs jump to form a bigger square?
Rule: Click a peg P then click peg Q to jump P over Q. The jumper reflects through Q landing at P' = 2Q - P.
Lattice Covolume det(Λ) 1.00
Can Form Larger Square? NO
Centroid (x̄, ȳ) (0.50, 0.50)
Preserved Parity Classes 4 / 4

The Affine Reflection Invariant

When peg \(P\) reflects across peg \(Q\), the transformation in vector coordinates is:
P' = 2Q - P = P + 2(Q - P) Notice that \(P' \equiv P \pmod 2\). Every jump changes coordinates by a multiple of 2!

Consequence: The four pegs initially occupy distinct mod-2 parity classes: \((0,0), (1,0), (1,1), (0,1)\). No jump can ever alter a peg's parity class mod 2. Furthermore, the basis vectors spanning the peg lattice transform via matrices with integer entries and determinant = ±1. Hence, Lattice Covolume is strictly conserved = 1.

PegColorCurrent (x, y)Parity (x mod 2, y mod 2)