Four Square Entries in a 3×3 Magic Square — 23 Orbits

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The level is a lower guarantee, not a prohibition on additional square-valued cells. Open a mask to load the same laboratory with its parameters and complete proof directly below the square.

Proof of classification completeness and sufficiency of the quadrics →

Proof table

All 23 orbits and quadrics for 4/9

Each cell uses the color of the quadric containing its square value. An intersection of two supports is split between both colors.

Upper block: original system; colored bars: quadrics:Light-red arithmetic progressionGreen quadric relationYellow equality of two sums of squaresDark-gray quadric relationBlue x² + 2y² normBrown quadric relationLight-gray quadric relation

01

The center and three corners

\left\{\begin{aligned}E+x&=a^2\\E-y&=c^2\\E&=e^2\\E+y&=g^2\end{aligned}\right.

c^2+g^2=2e^2

02

The center and three edge cells

\left\{\begin{aligned}E-x+y&=b^2\\E-x-y&=d^2\\E&=e^2\\E+x+y&=f^2\end{aligned}\right.

d^2+f^2=2e^2

03

The center, two adjacent corners, and the edge between them

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-y&=c^2\\E&=e^2\end{aligned}\right.

a^2+b^2+c^2=3e^2

04

The center, two adjacent corners, and the opposite edge

\left\{\begin{aligned}E+x&=a^2\\E-y&=c^2\\E&=e^2\\E+x-y&=h^2\end{aligned}\right.

a^2+c^2=e^2+h^2

05

The center, two adjacent corners, and an edge incident to one of them

\left\{\begin{aligned}E+x&=a^2\\E-y&=c^2\\E-x-y&=d^2\\E&=e^2\end{aligned}\right.

a^2+d^2=c^2+e^2

06

The center, two opposite corners, and one edge

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E&=e^2\\E-x&=j^2\end{aligned}\right.

a^2+j^2=2e^2

07

The center, two adjacent edges, and the corner between them

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-x-y&=d^2\\E&=e^2\end{aligned}\right.

2a^2+b^2+d^2=4e^2

08

The center, two adjacent edges, and the opposite corner

\left\{\begin{aligned}E-x+y&=b^2\\E-x-y&=d^2\\E&=e^2\\E-x&=j^2\end{aligned}\right.

b^2+d^2=2j^2

09

The center, two adjacent edges, and a corner incident to one of them

\left\{\begin{aligned}E-x+y&=b^2\\E-y&=c^2\\E-x-y&=d^2\\E&=e^2\end{aligned}\right.

b^2+2c^2=d^2+2e^2

10

The center, two opposite edges, and one corner

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E&=e^2\\E+x-y&=h^2\end{aligned}\right.

b^2+h^2=2e^2

11

All four edge cells

\left\{\begin{aligned}E-x+y&=b^2\\E-x-y&=d^2\\E+x+y&=f^2\\E+x-y&=h^2\end{aligned}\right.

b^2+h^2=d^2+f^2

12

All four corners

\left\{\begin{aligned}E+x&=a^2\\E-y&=c^2\\E+y&=g^2\\E-x&=j^2\end{aligned}\right.

a^2+j^2=c^2+g^2

13

Three edge cells and an outer corner

\left\{\begin{aligned}E-x+y&=b^2\\E-x-y&=d^2\\E+x+y&=f^2\\E+y&=g^2\end{aligned}\right.

b^2+f^2=2g^2

14

Three corners and the inner edge

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-y&=c^2\\E+y&=g^2\end{aligned}\right.

2a^2+2b^2=c^2+3g^2

15

Three corners and the outer edge

\left\{\begin{aligned}E+x&=a^2\\E-y&=c^2\\E+x+y&=f^2\\E+y&=g^2\end{aligned}\right.

2a^2+g^2=c^2+2f^2

16

Three edge cells and the inner corner

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-x-y&=d^2\\E+x+y&=f^2\end{aligned}\right.

2a^2+b^2=d^2+2f^2

17

Two adjacent corners and the two noncommon incident edges

\left\{\begin{aligned}E+x&=a^2\\E-y&=c^2\\E-x-y&=d^2\\E+x+y&=f^2\end{aligned}\right.

2a^2+d^2=2c^2+f^2

18

Two adjacent corners, their common edge, and the opposite edge

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-y&=c^2\\E+x-y&=h^2\end{aligned}\right.

2a^2+2c^2=b^2+3h^2

19

Two adjacent corners and two edges incident to one corner

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-y&=c^2\\E-x-y&=d^2\end{aligned}\right.

2a^2+3d^2=b^2+4c^2

20

Two opposite corners and two edges incident to one of them

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-x-y&=d^2\\E-x&=j^2\end{aligned}\right.

b^2+d^2=2j^2

21

Two opposite corners and adjacent edges incident to them

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E+x+y&=f^2\\E-x&=j^2\end{aligned}\right.

a^2+b^2=f^2+j^2

22

Two opposite corners and two opposite edges

\left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E+x-y&=h^2\\E-x&=j^2\end{aligned}\right.

a^2+j^2=b^2+h^2

23

Two adjacent corners and two adjacent edges at the unselected corner

\left\{\begin{aligned}E+x&=a^2\\E-y&=c^2\\E-x-y&=d^2\\E+x-y&=h^2\end{aligned}\right.

2c^2=d^2+h^2