11 parametric solutions of 4x4 magic squares of squares
by Seiji Tomita, November 2018.


Solution #1, 4x4 magic square of squares, S = 50(k²+337)

(2k+105)²

(3k+38)²

(6k-59)²

(k+30)²

(k-66)²

(6k+5)²

(3k+70)²

(2k-87)²

(3k-38)²

(2k-105)²

(k-30)²

(6k+59)²

(6k-5)²

(k+66)²

(2k+87)²

(3k-70)²

 

Solution #2, 4x4 magic square of squares, S = 130(k²+281)

(3k+154)²

(6k+13)²

(9k-78)²

(2k+81)²

(2k-111)²

(9k+18)²

(6k+77)²

(3k-134)²

(6k-13)²

(3k-154)²

(2k-81)²

(9k+78)²

(9k-18)²

(2k+111)²

(3k+134)²

(6k-77)²

 

Solution #3, 4x4 magic square of squares, S = 125(k²+373)

(4k+165)²

(6k+2)²

(8k-114)²

(3k+80)²

(3k-136)²

(8k+30)²

(6k+110)²

(4k-123)²

(6k-2)²

(4k-165)²

(3k-80)²

(8k+114)²

(8k-30)²

(3k+136)²

(4k+123)²

(6k-110)²

 

Solution #4, 4x4 magic square of squares, S = 130(2k²+53)

(3k+70)²

(5k+33)²

(15k-26)²

(k+15)²

(k-30)²

(15k+1)²

(5k+42)²

(3k-65)²

(5k-33)²

(3k-70)²

(k-15)²

(15k+26)²

(15k-1)²

(k+30)²

(3k+65)²

(5k-42)²

 

Solution #5, 4x4 magic square of squares, S = 145(k²+74)

(4k+80)²

(5k+36)²

(10k-53)²

(2k+15)²

(2k-55)²

(10k+3)²

(5k+64)²

(4k-60)²

(5k-36)²

(4k-80)²

(2k-15)²

(10k+53)²

(10k-3)²

(2k+55)²

(4k+60)²

(5k-64)²

 

Solution #6, 4x4 magic square of squares, S = 340(k²+29)

(5k+81)²

(9k+15)²

(15k-43)²

(3k+35)²

(3k-55)²

(15k+7)²

(9k+45)²

(5k-69)²

(9k-15)²

(5k-81)²

(3k-35)²

(15k+43)²

(15k-7)²

(3k+55)²

(5k+69)²

(9k-45)²

 

Solution #7, 4x4 magic square of squares, S = 290(2k²+37)

(7k+81)²

(9k+42)²

(21k-47)²

(3k+14)²

(3k-49)²

(21k+2)²

(9k+63)²

(7k-66)²

(9k-42)²

(7k-81)²

(3k-14)²

(21k+47)²

(21k-2)²

(3k+49)²

(7k+66)²

(9k-63)²

 

Solution #8, 4x4 magic square of squares, S = 130(k²+281)

(2k+165)²

(5k+34)²

(10k-57)²

(k+70)²

(k-90)²

(10k+7)²

(5k+66)²

(2k-155)²

(5k-34)²

(2k-165)²

(k-70)²

(10k+57)²

(10k-7)²

(k+90)²

(2k+155)²

(5k-66)²

 

Solution #9, 4x4 magic square of squares, S = 221(k²+85)

(3k+112)²

(8k+6)²

(12k-43)²

(2k+66)²

(2k-78)²

(12k+11)²

(8k+42)²

(3k-104)²

(8k-6)²

(3k-112)²

(2k-66)²

(12k+43)²

(12k-11)²

(2k+78)²

(3k+104)²

(8k-42)²

 

Solution #10, 4x4 magic square of squares, S = 481(k²+50)

(3k+128)²

(12k+4)²

(18k-33)²

(2k+81)²

(2k-87)²

(18k+9)²

(12k+32)²

(3k-124)²

(12k-4)²

(3k-128)²

(2k-81)²

(18k+33)²

(18k-9)²

(2k+87)²

(3k+124)²

(12k-32)²

 

Solution #11, 4x4 magic square of squares, S = 325(k²+149)

(8k+162)²

(9k+24)²

(12k-143)²

(6k+34)²

(6k-146)²

(12k+17)²

(9k+144)²

(8k-78)²

(9k-24)²

(8k-162)²

(6k-34)²

(12k+143)²

(12k-17)²

(6k+146)²

(8k+78)²

(9k-144)²


Construction method

S = (a1² + a2² + a3² + a4²)k² + (b1² + b2² + b3² + b4²)

(a1k + b1

(a2k + b2

(a3k - b3

(a4k + b4

(a4k - c1

(a3k + c2

(a2k + c3

(a1k - c4

(a2k - b2

(a1k - b1

(a4k - b4

(a3k + b3

(a3k - c2

(a4k + c1

(a1k + c4

(a2k - c3

And solving simultaneous equations:

  1. b1² + b2² = c3² + c4²
  2. b3² + b4² = c1² + c2²
  3. b2² + b3² = c2² + c3²
  4. b1² + b4² = c1² + c4²
  5. a1b1 + a2b2 + a4b4 = a3b3
  6. a1c4 + a2c3 = a3b3 + a4b4
  7. a1c4 + a4c1 = a2c3 + a3c2
  8. a1b1 + a3c2 = a2c3 + a4b4
  9. a2b2 + a3c2 + a4c1 = a1b1

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