Five 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 pairs of quadrics for 5/9

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 progressionDark-red arithmetic progressionYellow equality of two sums of squaresBlue x² + 2y² normBlue x² + 2y² normWeighted brown conic

01

Complement of the 4/9 type “All four edge cells”

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

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

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

02

Complement of the 4/9 type “All four corners”

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

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

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

03

Complement of the 4/9 type “Two opposite corners and two opposite edges”

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

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

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

04

Complement of the 4/9 type “Three corners and the outer edge”

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

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

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

05

Complement of the 4/9 type “The center, two adjacent corners, and the opposite edge”

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

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

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

06

Complement of the 4/9 type “Two opposite corners and adjacent edges incident to them”

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

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

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

07

Complement of the 4/9 type “The center and three corners”

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

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

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

08

Complement of the 4/9 type “The center, two opposite corners, and one edge”

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

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

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

09

Complement of the 4/9 type “Three edge cells and an outer corner”

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

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

c^2+e^2=h^2+j^2

10

Complement of the 4/9 type “Three edge cells and the inner corner”

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

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

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

11

Complement of the 4/9 type “Two adjacent corners and the two noncommon incident edges”

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

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

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

12

Complement of the 4/9 type “Two adjacent corners, their common edge, and the opposite edge”

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

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

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

13

Complement of the 4/9 type “Two opposite corners and two edges incident to one of them”

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

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

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

14

Complement of the 4/9 type “The center, two adjacent edges, and the opposite corner”

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

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

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

15

Complement of the 4/9 type “Two adjacent corners and two adjacent edges at the unselected corner”

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

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

b^2+e^2=g^2+j^2

16

Complement of the 4/9 type “The center, two adjacent corners, and an edge incident to one of them”

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

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

b^2+2h^2=d^2+2a^2

17

Complement of the 4/9 type “The center, two adjacent edges, and a corner incident to one of them”

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

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

a^2+2d^2=j^2+2c^2

18

Complement of the 4/9 type “Three corners and the inner edge”

\left\{\begin{aligned}E+x&=a^2\\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

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

19

Complement of the 4/9 type “The center and three edge cells”

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

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

c^2+2b^2=g^2+2j^2

20

Complement of the 4/9 type “The center, two opposite edges, and one corner”

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

b^2+h^2=c^2+g^2

g^2+2h^2=c^2+2a^2

21

Complement of the 4/9 type “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\\E&=e^2\end{aligned}\right.

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

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

22

Complement of the 4/9 type “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-x-y&=d^2\\E+x+y&=f^2\end{aligned}\right.

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

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

23

Complement of the 4/9 type “The center, two adjacent edges, and the corner between them”

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

b^2+c^2=d^2+g^2

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