Partitioning| (x+y+z)^4 – (y+z)^4 – (z+x)^4 – (x+y)^4 + x^4 + y^4 + z^4

Consider the identity:

$(x+y+z)^4 - (y+z)^4 - (z+x)^4 - (x+y)^4 + x^4 + y^4 + z^4 = 12 \; x \; y \; z \; (x+y+z)$

118 can be partitioned into three parts that have the same product in 4 different ways:

$14 + 50 + 54 = 15 + 40 + 63 = 18 + 30 + 70 = 21 + 25 + 72 = 118$     and
$14 \times 50 \times 54 = 15 \times 40 \times 63 = 18 \times 30 \times 70 = 21 \times 25 \times 72 = 37800$

It is the smallest number for which that is possible.

The next smallest one is 130

$15 + 24 + 91 = 14 + 26 + 90 = 10 + 42 + 78 = 9 + 56 + 65 = 130$
$15 \times 24 \times 91 = 14 \times 26 \times 90 = 10 \times 42 \times 78 = 9 \times 56 \times 65 = 32760$

(1)

$x+y+z = 118$        $x\cdot y\cdot z = 37800$

$12 \; x \; y \; z \; (x+y+z) = (x+y+z)^4 - (y+z)^4 - (z+x)^4 - (x+y)^4 + x^4 + y^4 + z^4$

$12 \times 37800 \times 118 = 118^4 - (50+54)^4 - (54+14)^4 - (14+50)^4 + 14^4 + 50^4 + 54^4$
$12 \times 37800 \times 118 = 118^4 - (40+63)^4 - (63+15)^4 - (15+40)^4 + 15^4 + 40^4 + 63^3$
$12 \times 37800 \times 118 = 118^4 - (30+70)^4 - (70+18)^4 - (18+30)^4 + 18^4 + 30^4 + 70^4$
$12 \times 37800 \times 118 = 118^4 - (25+72)^4 - (72+21)^4 - (21+25)^4 + 21^4 + 25^4 + 72^4$

$12 \times 37800 \times 118 = 118^4 - 104^4 - 68^4 - 64^4 + 14^4 + 50^4 + 54^4 = 53524800$
$12 \times 37800 \times 118 = 118^4 - 103^4 - 78^4 - 55^4 + 15^4 + 40^4 + 63^4 = 53524800$
$12 \times 37800 \times 118 = 118^4 - 100^4 - 88^4 - 48^4 + 18^4 + 30^4 + 70^4 = 53524800$
$12 \times 37800 \times 118 = 118^4 - 97^4 - 93^4 - 46^4 + 21^4 + 25^4 + 72^4 = 53524800$

(2)

$x+y+z = 130$              $x\cdot y\cdot z = 32760$
$12 \; x \; y \; z \; (x+y+z) \; = \; 12 \times 32760 \times 130 \; = \; 51105600$

$51105600 = 130^4 - (24+91)^4 - (91+15)^4 - (15+24)^4 + 15^4 + 24^4 + 91^4$
$51105600 = 130^4 - (26+90)^4 - (90+14)^4 - (14+26)^4 + 14^4 + 26^4 + 90^4$
$51105600 = 130^4 - (42+78)^4 - (78+10)^4 - (10+42)^4 + 10^4 + 42^4 + 78^4$
$51105600 = 130^4 - (56+65)^4 - (65+9)^4 - (9+56)^4 + 9^4 + 56^4 + 65^4$

$51105600 = 130^4 - 115^4 - 106^4 - 39^4 + 15^4 + 24^4 + 91^4$
$51105600 = 130^4 - 116^4 - 104^4 - 40^4 + 14^4 + 26^4 + 90^4$
$51105600 = 130^4 - 120^4 - 88^4 - 52^4 + 10^4 + 42^4 + 78^4$
$51105600 = 130^4 - 121^4 - 74^4 - 65^4 + 9^4 + 56^4 + 65^4$

(1)

$103^4 + 78^4 + 55^4 + 54^4 + 50^4 + 14^4 = 104^4 + 68^4 + 64^4 + 63^4 + 40^4 + 15^4 = 173508034$
$100^4 + 88^4 + 54^4 + 50^4 + 48^4 + 14^4 = 104^4 + 70^4 + 68^4 + 64^4 + 30^4 + 18^4 = 180069424$
$97^4 + 93^4 + 54^4 + 50^4 + 46^4 + 14^4 = 104^4 + 72^4 + 68^4 + 64^4 + 25^4 + 21^4 = 182603410$
$100^4 + 88^4 + 63^4 + 48^4 + 40^4 + 15^4 = 103^4 + 78^4 + 70^4 + 55^4 + 30^4 + 18^4 = 183641538$
$97^4 + 93^4 + 63^4 + 46^4 + 40^4 + 15^4 = 103^4 + 78^4 + 72^4 + 55^4 + 25^4 + 21^4 = 186175524$
$97^4 + 93^4 + 70^4 + 46^4 + 30^4 + 18^4 = 100^4 + 88^4 + 72^4 + 48^4 + 25^4 + 21^4 = 192736914$

(2)

$116^4 + 104^4 + 91^4 + 40^4 + 24^4 + 15^4$
$= \; 115^4 + 106^4 + 90^4 + 39^4 + 26^4 + 14^4 = 369567154$

$120^4 + 91^4 + 88^4 + 52^4 + 24^4 + 15^4$
$= \; 115^4 + 106^4 + 78^4 + 42^4 + 39^4 + 10^4 = 343598514$

$121^4 + 91^4 + 74^4 + 65^4 + 24^4 + 15^4$
$= \; 115^4 + 106^4 + 65^4 + 56^4 + 39^4 + 9^4 = 331153444$

$120^4 + 88^4 + 90^4 + 52^4 + 26^4 + 14^4$
$= \; 116^4 + 104^4 + 78^4 + 40^4 + 42^4 + 10^4 = 340746544$

$121^4 + 90^4 + 74^4 + 65^4 + 26^4 + 14^4$
$= \; 116^4 + 104^4 + 65^4 + 56^4 + 40^4 + 9^4 = 328301474$

$121^4 + 78^4 + 74^4 + 65^4 + 42^4 + 10^4$
$= \; 120^4 + 88^4 + 65^4 + 56^4 + 52^4 + 9^4 = 302332834$