Magic Cube — Order 3×3

Created by Inder J. Taneja

Six faces · Numbers 1–54 · Pink checkerboard coloring

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19
49
7
13
25
37
43
1
31
20
50
8
14
26
38
44
2
32
21
51
9
15
27
39
45
3
33
22
52
10
16
28
40
46
4
34
23
53
11
17
29
41
47
5
35
24
54
12
18
30
42
48
6
36
Magic Constant
75
Face 1 · S = 3f + 72
Numbers Used
1 – 54
9 per face × 6 faces
MC Increment
+3
Face f: S = 3f + 72
Faces
6

Data Analysis — 3×3 Magic Cube

Six faces · Numbers 1–54 · Magic constants 75 to 90

This analysis covers the 3×3 magic cube using numbers 1–54 across six faces. The magic constant for face f follows the formula S = 3f + 72, producing constants from 75 (face 1) to 90 (face 6). Every face is a valid magic square — rows, columns, and both main diagonals all sum to the face magic constant.
Cube Overview
3×3

Magic Cube — Uniform Pink Checkerboard

Alternating pink checkerboard · 6 faces · Numbers 1–54
Magic Constant Range75 → 90
Face MC FormulaS = 3f + 72
MC Increment (face to face)+3
Both Diagonals= Face MC ✓ on every face
Numbers Used1 – 54 (consecutive integers)
Face Total (Face 1)225 = 3 × 75
Grand Total (all 6 faces)1,485 = sum(1…54)
Centre cell (Face f)= MC / 3 (always the median)
Magic Constant Progression
FaceFormulaMagic ConstantFace TotalIncrementCentre CellNote
13×1 + 72 75 225 — 25 Min MC · = 3×25
23×2 + 72 78 234 +3 26
33×3 + 72 81 243 +3 27 = 3×27
43×4 + 72 84 252 +3 28 = 3×28
53×5 + 72 87 261 +3 29
63×6 + 72 90 270 +3 30 Max MC · = 3×30
Row, Column & Diagonal Sums — All Six Faces
FaceMC R1R2R3 C1C2C3 D↘D↙ Rows ✓Cols ✓Diags ✓
F1 75 757575 757575 7575 ✓ ✓ ✓
F2 78 787878 787878 7878 ✓ ✓ ✓
F3 81 818181 818181 8181 ✓ ✓ ✓
F4 84 848484 848484 8484 ✓ ✓ ✓
F5 87 878787 878787 8787 ✓ ✓ ✓
F6 90 909090 909090 9090 ✓ ✓ ✓
Face Grids — Full Verification
Grid Font Size 16px — drag to resize numbers

Face 1  ·  S = 75

19
49
7
13
25
37
43
1
31
R1:75R2:75R3:75
C1:75C2:75C3:75
D↘75 D↙75 ✓
Centre: 25 = MC/3

Face 2  ·  S = 78

20
50
8
14
26
38
44
2
32
R1:78R2:78R3:78
C1:78C2:78C3:78
D↘78 D↙78 ✓
Centre: 26 = MC/3

Face 3  ·  S = 81

21
51
9
15
27
39
45
3
33
R1:81R2:81R3:81
C1:81C2:81C3:81
D↘81 D↙81 ✓
Centre: 27 = MC/3

Face 4  ·  S = 84

22
52
10
16
28
40
46
4
34
R1:84R2:84R3:84
C1:84C2:84C3:84
D↘84 D↙84 ✓
Centre: 28 = MC/3

Face 5  ·  S = 87

23
53
11
17
29
41
47
5
35
R1:87R2:87R3:87
C1:87C2:87C3:87
D↘87 D↙87 ✓
Centre: 29 = MC/3

Face 6  ·  S = 90

24
54
12
18
30
42
48
6
36
R1:90R2:90R3:90
C1:90C2:90C3:90
D↘90 D↙90 ✓
Centre: 30 = MC/3
Special Properties
PropertyValue / CheckNote
Numbers partition Residue classes mod 6 Faces use 6 distinct residue classes of {1…54}
Grand total 1,485 sum(1…54) = 54×55/2 = 1,485
Sum of all MCs sum(75,78,81,84,87,90) = 495 = 6×82.5 (mean MC)
Face total progression 225, 234, 243, 252, 261, 270 Arithmetic sequence, step +9
Centre cell progression 25, 26, 27, 28, 29, 30 Consecutive integers, = MC/3
All faces magic? ✓ Yes — all 6 faces Rows, cols, diagonals verified
Pandiagonal? Not verified (3×3 limit) No pandiagonal 3×3 magic square exists
Colour scheme Alternating checkerboard No sub-block constraint on 3×3

Key Findings

1
Cube structure (uniform pink)
6
Magic squares (one per face)
75
Face 1 magic constant (= 3 × 25)
225
Face 1 total = 3 × 75
+3
MC increment face to face
54
Numbers used (1–54)
1,485
Grand total = sum(1…54)
100%
Rows, cols & diagonals magic ✓
3×3 Magic Cube Research · Data Analysis — Uniform Pink Checkerboard