## Sums & Products| x+y+z=a+b+c and x*y*z=a*b*c

Note that

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.

Find the next smallest one.

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Find 6 distinct integers so that

$x \; + \; y \; + \; z \; = \; a \; + \; b \; + \; c$     and
$x \cdot y \cdot z \; = \; a \cdot b \cdot c$

Here are some solutions:

$3+4+12 = 2+8+9 = 19$ ……….. $3\times 4\times 12 = 2\times 8\times 9 = 144$
$3+4+14 = 2+7+12 = 21$ ………. $4\times 3\times 14 = 2\times 7\times 12 = 168$
$3+5+16 = 2+10+12 = 24$ ……… $3\times 5\times 16 = 2\times 10\times 12 = 240$
$3+5+18 = 2+15+9 = 26$ ………. $3\times 5\times 18 = 2\times 15\times 9 = 270$
$3+5+32 = 2+8+30 = 40$ ………. $3\times 5\times 32 = 2\times 8\times 30 = 480$
$3+6+20 = 2+12+15 = 29$ ……… $3\times 6\times 20 = 2\times 12\times 15 = 360$
$3+6+22 = 2+11+18 = 31$ ……… $3\times 6\times 22 = 2\times 11\times 18 = 396$
$3+6+30 = 2+10+27 = 39$ ……… $3\times 6\times 30 = 2\times 10\times 27 = 540$
$3+7+24 = 2+14+18 = 34$ ……… $3\times 7\times 24 = 2\times 14\times 18 = 504$
$3+7+26 = 2+13+21 = 36$ ……… $3\times 7\times 26 = 2\times 13\times 21 = 546$
$3+7+32 = 2+12+28 = 42$ ……… $3\times 7\times 32 = 2\times 12\times 28 = 672$
$3+8+28 = 2+16+21 = 39$ ……… $3\times 8\times 28 = 2\times 16\times 21 = 672$
$3+8+30 = 2+15+24 = 41$ ……… $3\times 8\times 30 = 2\times 15\times 24 = 720$
$3+8+35 = 2+14+30 = 46$ ……… $3\times 8\times 35 = 2\times 14\times 30 = 840$
$3+8+52 = 2+13+48 = 63$ ……… $3\times 8\times 52 = 2\times 13\times 48 = 1248$
$3+9+32 = 2+18+24 = 44$ ……… $3\times 9\times 32 = 2\times 18\times 24 = 864$
$3+9+34 = 2+17+27 = 46$ ……… $3\times 9\times 34 = 2\times 17\times 27 = 918$
$3+9+50 = 2+15+45 = 62$ ……… $3\times 9\times 50 = 2\times 15\times 45 = 1350$
$3+10+35 = 2+21+25 = 48$ …….. $3\times 10\times 35 = 2\times 21\times 25 = 1050$
$3+10+36 = 2+20+27 = 49$ …….. $3\times 10\times 36 = 2\times 20\times 27 = 1080$
$3+10+38 = 2+19+30 = 51$ …….. $3\times 10\times 38 = 2\times 19\times 30 = 1140$
$3+10+42 = 2+18+35 = 55$ …….. $3\times 10\times 42 = 2\times 18\times 35 = 1260$
$3+10+51 = 2+17+45 = 64$ …….. $3\times 10\times 51 = 2\times 17\times 45 = 1530$
$3+10+80 = 2+16+75 = 93$ …….. $3\times 10\times 80 = 2\times 16\times 75 = 2400$
$3+11+40 = 2+22+30 = 54$ …….. $3\times 11\times 40 = 2\times 22\times 30 = 1320$
$3+11+42 = 2+21+33 = 56$ …….. $3\times 11\times 42 = 2\times 21\times 33 = 1386$
$3+11+72 = 2+18+66 = 86$ …….. $3\times 11\times 72 = 2\times 18\times 66 = 2376$
$3+12+44 = 2+24+33 = 59$ …….. $3\times 12\times 44 = 2\times 24\times 33 = 1584$
$3+12+46 = 2+23+36 = 61$ …….. $3\times 12\times 46 = 2\times 23\times 36 = 1656$
$3+12+56 = 2+21+48 = 71$ …….. $3\times 12\times 56 = 2\times 21\times 48 = 2016$
$3+12+70 = 2+20+63 = 85$ …….. $3\times 12\times 70 = 2\times 20\times 63 = 2520$
$3+13+98 = 2+21+91 = 114$ ……. $3\times 13\times 98 = 2\times 21\times 91 = 3822$
$3+14+72 = 2+24+63 = 89$ …….. $3\times 14\times 72 = 2\times 24\times 63 = 3024$
$3+14+92 = 2+23+84 = 109$ ……. $3\times 14\times 92 = 2\times 23\times 84 = 3864$

$4+5+24 = 2+15+16 = 33$ ……… $4\times 5\times 24 = 2\times 15\times 16 = 480$
$4+5+26 = 2+13+20 = 35$ ……… $4\times 5\times 26 = 2\times 13\times 20 = 520$
$4+5+30 = 2+12+25 = 39$ ……… $4\times 5\times 30 = 2\times 12\times 25 = 600$
$4+5+44 = 2+11+40 = 53$ ……… $4\times 5\times 44 = 2\times 11\times 40 = 880$
$4+6+30 = 2+18+20 = 40$ ……… $4\times 6\times 30 = 2\times 18\times 20 = 720$
$4+6+32 = 2+16+24 = 42$ ……… $4\times 6\times 32 = 2\times 16\times 24 = 768$
$4+6+35 = 2+15+28 = 45$ ……… $4\times 6\times 35 = 2\times 15\times 28 = 840$
$4+6+42 = 2+14+36 = 52$ ……… $4\times 6\times 42 = 2\times 14\times 36 = 1008$
$4+6+65 = 2+13+60 = 75$ ……… $4\times 6\times 65 = 2\times 13\times 60 = 1560$
$4+7+36 = 2+24+21 = 47$ ……… $4\times 7\times 36 = 2\times 24\times 21 = 1008$
$4+7+38 = 2+19+28 = 49$ ……… $4\times 7\times 38 = 2\times 19\times 28 = 1064$
$4+7+56 = 2+16+49 = 67$ ……… $4\times 7\times 56 = 2\times 16\times 49 = 1568$
$4+7+90 = 2+15+84 = 101$ …….. $4\times 7\times 90 = 2\times 15\times 84 = 2520$
$4+8+42 = 2+24+28 = 54$ ……… $4\times 8\times 42 = 2\times 24\times 28 = 1344$
$4+8+44 = 2+22+32 = 56$ ……… $4\times 8\times 44 = 2\times 22\times 32 = 1408$
$4+8+50 = 2+20+40 = 62$ ……… $4\times 8\times 50 = 2\times 20\times 40 = 1600$
$4+8+57 = 2+19+48 = 69$ ……… $4\times 8\times 57 = 2\times 19\times 48 = 1824$
$4+8+72 = 2+18+64 = 84$ ……… $4\times 8\times 72 = 2\times 18\times 64 = 2304$
$4+9+52 = 2+24+39 = 65$ ……… $4\times 9\times 52 = 2\times 24\times 39 = 1872$
$4+9+70 = 2+21+60 = 83$ ……… $4\times 9\times 70 = 2\times 21\times 60 = 2520$
$4+9+90 = 2+20+81 = 103$ …….. $4\times 9\times 90 = 2\times 20\times 81 = 3240$
$4+10+72 = 2+24+60 = 86$ …….. $4\times 10\times 72 = 2\times 24\times 60 = 2880$

$5+6+40 = 2+24+25 = 51$ ……… $5\times 6\times 40 = 2\times 24\times 25 = 1200$
$5+6+42 = 2+21+30 = 53$ ……… $5\times 6\times 42 = 2\times 21\times 30 = 1260$
$5+6+44 = 2+20+33 = 55$ ……… $5\times 6\times 44 = 2\times 20\times 33 = 1320$
$5+6+54 = 2+18+45 = 65$ ……… $5\times 6\times 54 = 2\times 18\times 45 = 1620$
$5+6+68 = 2+17+60 = 79$ ……… $5\times 6\times 68 = 2\times 17\times 60 = 2040$
$5+7+66 = 2+21+55 = 78$ ……… $5\times 7\times 66 = 2\times 21\times 55 = 2310$
$5+7+80 = 2+20+70 = 92$ ……… $5\times 7\times 80 = 2\times 20\times 70 = 2800$
$5+8+78 = 2+24+65 = 91$ ……… $5\times 8\times 78 = 2\times 24\times 65 = 3120$
$5+8+92 = 2+23+80 = 105$ …….. $5\times 8\times 92 = 2\times 23\times 80 = 3680$