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123456 Triangle (Posted on 2023-10-28)

Find all integer-sided triangles (a,b,c), where each side is a permutation of N=123456, and the area is an integer.

List your triangles with a<b<c

No Solution Yet Submitted by K Sengupta

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Computer solution

Comment 1 of 1

I found 481 such triangles.

Since the problem states a<b<c, I did not account for any possible isosceles triangles.

I used Heron's formula to calculate area, and accounted for potential rounding errors. Hopefully this did not add any spurious results.

-----------------------

def areaHeron(a,b,c):

""" input 3 float numbers, the 3 sides of a triangle

output the area """

if c >= a+b or b >= a+c or a >= b+c:

return 0

s = (a+b+c)/2 # semiperimeter

return (s*(s-a)*(s-b)*(s-c))**(1/2)

triangles = []

perms1to6 = []

from itertools import permutations

from itertools import combinations

for p in permutations('123456'):

perms1to6.append(int(''.join(p)))

perms1to6 = sorted(perms1to6)

for comb in combinations(perms1to6,3):

if comb[2] < comb[1] or comb[1] < comb[0]:

continue

area = areaHeron(comb[0],comb[1],comb[2])

if area <= 0:

continue

if abs(round(area) - area) < .000001:

print(list(comb) + [area])

triangles.append(list(comb) + [area])

-----------------------

[123546, 436215, 451236, 26941974636.0]

[123564, 146235, 234615, 7777227827.000001]

[124356, 265431, 326154, 15691457777.0]

[124536, 135426, 152364, 8002629311.999999]

[124536, 163245, 235164, 9619161368.0]

[124563, 342516, 465231, 4265134651.000001]

[125346, 425316, 526413, 17477529162.0]

[125634, 451236, 546123, 20367824039.0]

[126435, 651423, 653241, 41040449995.0]

[126543, 541326, 635241, 24802092261.0]

[132564, 456312, 541632, 25088012410.0]

[132645, 546312, 654312, 22973813356.0]

[132654, 261345, 326154, 16560974829.0]

[134256, 513462, 561423, 33432967579.0]

[134526, 451236, 513246, 28504009008.0]

[134652, 521643, 635124, 20817876349.0]

[134652, 523146, 532416, 35159469985.0]

[135246, 421536, 453612, 28400483088.0]

[136245, 312564, 412653, 16461093706.0]

[136452, 513462, 612534, 26218119928.0]

[136542, 264531, 341256, 16665263816.0]

[142365, 253164, 256431, 17410386418.0]

[142536, 246513, 346512, 14617129347.0]

[143256, 214356, 254613, 15348396380.0]

[143526, 612354, 635412, 43896852684.0]

[143562, 312465, 341265, 22424622228.0]

[145236, 146235, 264513, 8095784025.0]

[145362, 436512, 532461, 26152796483.0]

[145623, 536124, 614235, 35060810846.0]

[145632, 413526, 531264, 20002864224.0]

[146253, 352614, 415362, 24899830716.0]

[146253, 634152, 653412, 46365486332.0]

[152346, 431562, 536412, 26413000488.0]

[152346, 561342, 562341, 42401084558.0]

[152364, 632415, 654321, 48162599370.0]

[152436, 615423, 621543, 46742507231.0]

[152463, 263415, 351462, 18535865115.0]

[153246, 256143, 314652, 19469372898.0]

[153264, 615432, 651432, 46836531352.0]

[153426, 352164, 462351, 21350873607.0]

[153624, 431652, 512436, 30429614166.0]

[153624, 526134, 526341, 39988398832.0]

[153624, 531642, 613542, 36874133253.0]

[154236, 254316, 326154, 19094332956.0]

[154326, 461325, 531624, 33689582771.0]

[154623, 416352, 524631, 25613658765.0]

[154632, 532146, 541362, 40993884244.0]

[156234, 342516, 345216, 26155558062.0]

[156234, 431256, 453612, 33668559992.0]

[156342, 521364, 542316, 40751978922.0]

[156432, 516234, 536421, 40369728111.0]

[156432, 541632, 546213, 42083222866.0]

[162435, 234516, 312546, 18605665032.0]

[162435, 312645, 361452, 25339946097.0]

[162435, 643251, 653142, 52133094846.0]

[162453, 324156, 341562, 26068668215.0]

[162543, 312546, 326154, 25011409160.0]

[162543, 341625, 351246, 27321630972.0]

[162543, 421653, 465312, 34130043516.0]

[163452, 641523, 653214, 52349123427.0]

[164235, 231654, 325461, 17941587387.0]

[164235, 532461, 541236, 43503642060.0]

[164253, 251643, 324516, 20323146917.0]

[164352, 231654, 325614, 17950563639.0]

[164352, 416532, 532164, 27281641113.0]

[165243, 542361, 561342, 44782498514.0]

[165342, 245163, 324165, 19782272543.0]

[213456, 213465, 256143, 21871026171.0]

[213456, 425136, 564312, 39091372866.0]

[213456, 541326, 641532, 54819310598.0]

[213564, 243516, 246153, 23524570475.0]

[213564, 561324, 564312, 59001469192.0]

[213564, 624153, 642153, 66403771114.0]

[213645, 351624, 354261, 35931145271.0]

[213654, 516423, 654312, 46964204127.0]

[214356, 514632, 563412, 55132254690.0]

[214356, 516243, 641235, 49519776353.0]

[214365, 352641, 465312, 35988397051.0]

[214365, 532146, 532641, 55894754052.0]

[214536, 314265, 413526, 33067349045.0]

[214653, 235164, 413526, 18276755499.0]

[214653, 352641, 431652, 37638620793.0]

[215364, 341256, 415326, 36667713254.0]

[215436, 251463, 356214, 26741912598.0]

[215436, 513264, 536421, 55010889173.0]

[215463, 235146, 245631, 23123706797.0]

[215463, 365214, 426315, 39342066031.0]

[215634, 243561, 251436, 24003466019.0]

[215634, 245136, 415362, 20659484382.0]

[215634, 432156, 531264, 44955651007.0]

[215634, 465213, 531462, 49919649534.0]

[215643, 413652, 456123, 44536474250.0]

[216435, 531462, 653214, 52105368780.0]

[216435, 532164, 543162, 56920170344.0]

[216534, 362145, 542631, 26273638611.0]

[216543, 524163, 541632, 56310046951.0]

[231456, 452136, 462513, 51151154894.0]

[231456, 613542, 614325, 69775175470.0]

[231465, 234156, 346125, 26949245713.0]

[231546, 435621, 512643, 50198832268.0]

[231564, 326514, 421563, 37550959068.0]

[231564, 521463, 632154, 57465973140.0]

[231645, 425361, 614325, 33950220525.0]

[231645, 541632, 543612, 61397205900.0]

[231654, 452631, 623145, 41183031700.0]

[231654, 613542, 652341, 71055630612.0]

[234156, 352614, 436251, 41189139095.0]

[234156, 425163, 453216, 49201791552.0]

[234165, 412635, 624135, 25376676008.0]

[234516, 321546, 415623, 37532709318.0]

[234516, 324165, 436152, 37256216453.0]

[234561, 415362, 451263, 48345852213.0]

[234651, 243615, 451236, 17877476903.0]

[235146, 461325, 543126, 53821732956.0]

[235164, 354126, 461253, 40860801987.0]

[235461, 465321, 534162, 54680508101.0]

[235461, 625143, 632154, 72669419413.0]

[235614, 431562, 456123, 50137616520.0]

[235641, 513246, 621435, 58087948970.0]

[235641, 631524, 653124, 74081686154.0]

[236154, 436125, 452613, 50460067812.0]

[241356, 361254, 432651, 43593642652.0]

[241365, 316245, 534162, 21155447745.0]

[241365, 632154, 643251, 75488943437.0]

[241536, 312465, 415326, 37509041879.0]

[241635, 354216, 456123, 42365983666.0]

[241653, 431562, 641532, 31266408346.0]

[241653, 624531, 652341, 75242668320.0]

[243156, 245163, 316245, 29416625785.0]

[243165, 436215, 461532, 52249541982.0]

[243165, 512346, 526413, 61290059594.0]

[245136, 451632, 642153, 41106438851.0]

[245163, 251643, 416235, 28220225901.0]

[245613, 513264, 615243, 61527622747.0]

[245631, 321564, 463125, 37399843650.0]

[245631, 341562, 436521, 41812081104.0]

[245631, 351246, 431562, 43133788739.0]

[246513, 431652, 436251, 51275516386.0]

[246513, 624315, 652143, 76691582350.0]

[246513, 634152, 653241, 77637855538.0]

[246531, 264351, 356214, 32571720754.0]

[251346, 413256, 432561, 50597388758.0]

[251364, 316542, 321465, 36843114570.0]

[251364, 362154, 435162, 45499253701.0]

[251436, 465213, 623415, 51749074932.0]

[251463, 415326, 634152, 31558173728.0]

[251634, 362541, 521463, 41333270733.0]

[251634, 462153, 561234, 57362145602.0]

[251634, 513624, 514236, 62693319426.0]

[253164, 415236, 514263, 52094757741.0]

[253164, 536241, 645213, 65925952842.0]

[253461, 612345, 625143, 76653452690.0]

[253461, 632154, 642135, 79070884269.0]

[253614, 413562, 642153, 28142616572.0]

[253614, 562413, 642531, 70861626322.0]

[254136, 315462, 541326, 23829469231.0]

[254136, 341652, 564123, 26700895740.0]

[254136, 352614, 532614, 38031641267.0]

[254136, 361425, 542631, 38648142945.0]

[254136, 613425, 642351, 77625833700.0]

[254136, 652314, 654231, 81422267177.0]

[254163, 432516, 523614, 54676128672.0]

[254163, 562134, 643512, 70943785709.0]

[254361, 426531, 461325, 53583959439.0]

[254361, 641532, 645123, 80195966156.0]

[254613, 314625, 562134, 12526939496.0]

[254613, 462351, 524136, 58852303938.0]

[256134, 416325, 624351, 37681552710.0]

[256314, 561432, 632451, 71792962584.0]

[256341, 345126, 451362, 43981761320.0]

[256413, 345126, 452163, 43975472641.0]

[256413, 432516, 631254, 41817824577.0]

[256431, 261453, 463512, 26766112212.0]

[256431, 612345, 634251, 77921486434.0]

[261354, 415632, 451362, 53505828642.0]

[261354, 421635, 532146, 54311838418.0]

[261354, 543261, 645312, 69744772966.0]

[261435, 431562, 465321, 55604035151.0]

[261435, 641532, 653142, 82793353395.0]

[261534, 312564, 365421, 40054110517.0]

[261534, 531642, 654321, 66796301210.0]

[261543, 325164, 534126, 32185903733.0]

[263154, 452163, 564132, 58442776768.0]

[263451, 435612, 513624, 57369455057.0]

[263541, 431625, 624513, 45914757234.0]

[263541, 524613, 534162, 67517759570.0]

[264315, 461352, 562134, 60400568564.0]

[264315, 463251, 631452, 54148640860.0]

[264531, 653412, 654321, 84695621463.0]

[265134, 356214, 465312, 46974282169.0]

[265143, 341526, 541326, 36694532844.0]

[265143, 521364, 561423, 68789444951.0]

[265431, 412635, 534612, 53590397190.0]

[312456, 415263, 613254, 59215679881.0]

[312546, 421356, 632514, 57979135010.0]

[312546, 463152, 516432, 71479000206.0]

[312564, 356421, 623415, 37729226430.0]

[312645, 352164, 453261, 54899742727.0]

[312645, 456231, 634152, 67136862895.0]

[312654, 356412, 623145, 37855422943.0]

[314526, 325416, 513246, 49034224334.0]

[314625, 315462, 521634, 46089263501.0]

[314652, 426531, 564123, 66453338349.0]

[315246, 345261, 421356, 53444958590.0]

[315264, 315426, 452316, 49700772426.0]

[315264, 512436, 641235, 79834333323.0]

[315264, 562314, 654213, 88584400547.0]

[315426, 364521, 461253, 57280183389.0]

[315462, 536124, 625143, 84557872513.0]

[315462, 562341, 625314, 88482947141.0]

[315462, 615324, 625341, 94582119198.0]

[315642, 342516, 431652, 53511323407.0]

[315642, 356241, 362415, 50937146015.0]

[316245, 541632, 621435, 85625316397.0]

[316425, 324156, 512364, 49243200587.0]

[316425, 352614, 631452, 34844838825.0]

[316425, 543261, 612345, 85800856813.0]

[316452, 361452, 463521, 57052140654.0]

[316452, 546123, 561432, 83870964714.0]

[316524, 364215, 435612, 56626192031.0]

[316542, 451623, 465213, 68030056581.0]

[321456, 524631, 653214, 83460456820.0]

[321465, 415326, 653421, 55036526590.0]

[321465, 415632, 543261, 66635725832.0]

[321465, 435612, 456213, 66717062298.0]

[321546, 413562, 653124, 54535306104.0]

[321546, 462513, 643251, 70340573222.0]

[321654, 361542, 463215, 57938761151.0]

[324165, 462315, 645321, 70847461705.0]

[324165, 561432, 632541, 90858335601.0]

[324516, 624351, 654213, 99906159904.0]

[324615, 436215, 453261, 67112807672.0]

[324651, 612354, 652413, 98454364625.0]

[325146, 462135, 653142, 70178057526.0]

[325164, 345126, 436512, 55452149837.0]

[325416, 426531, 614235, 65697515045.0]

[325416, 436125, 631524, 66151186965.0]

[325614, 436251, 614532, 68053665648.0]

[326154, 365412, 435612, 58256744355.0]

[326154, 546312, 613542, 88807569211.0]

[326451, 346512, 416253, 54963051115.0]

[326451, 432651, 542613, 70619476133.0]

[326514, 361245, 412653, 56559949097.0]

[326514, 532641, 624513, 86957339104.0]

[341256, 412365, 653412, 60974116638.0]

[341256, 413562, 532614, 70558823047.0]

[341265, 341526, 641523, 37493363309.0]

[341265, 415632, 564231, 70545048590.0]

[341526, 541263, 614235, 92073818657.0]

[341526, 563142, 624153, 95526784064.0]

[341562, 435162, 543612, 74270881188.0]

[341625, 543216, 621354, 92565188274.0]

[341652, 516342, 651423, 87606562327.0]

[341652, 521364, 651432, 88611254541.0]

[342165, 365124, 615324, 53612865958.0]

[342615, 345216, 642153, 39568693451.0]

[342615, 423561, 516324, 72165346519.0]

[342651, 452613, 634152, 74928558267.0]

[342651, 465213, 536241, 78858661133.0]

[345126, 365124, 641325, 48911997357.0]

[345162, 361542, 653241, 44021435786.0]

[345162, 416253, 423651, 66057763575.0]

[345216, 463251, 632145, 78248773841.0]

[345261, 423651, 462153, 69972248771.0]

[345612, 416523, 562314, 71742436280.0]

[345612, 461352, 513264, 77843944089.0]

[345621, 564123, 635421, 97116662687.0]

[346251, 425631, 514623, 73129272530.0]

[346251, 523164, 614532, 90485313084.0]

[346521, 351264, 652143, 40467744121.0]

[346521, 536241, 563412, 90131416046.0]

[346521, 562413, 563142, 92770933982.0]

[351264, 612345, 614325, 103208609769.0]

[351426, 512346, 641532, 89798703757.0]

[351462, 354261, 514623, 62129035503.0]

[351642, 431562, 452361, 71167712473.0]

[351642, 546213, 625314, 95722837098.0]

[351642, 561243, 653142, 98632301092.0]

[352164, 435216, 452613, 71665584585.0]

[352416, 512436, 652431, 89770675049.0]

[352416, 564132, 621435, 98404477269.0]

[352461, 426153, 451263, 70621678904.0]

[352614, 642531, 653241, 109865371370.0]

[352641, 364125, 512436, 64187173437.0]

[354162, 523146, 564312, 90417656686.0]

[354216, 432516, 524136, 76016272123.0]

[354216, 521436, 642315, 92272231509.0]

[354612, 416523, 513642, 73319451503.0]

[354612, 421653, 625341, 71488936195.0]

[354612, 453216, 632415, 78495882272.0]

[354612, 523164, 564312, 90522672835.0]

[354621, 534126, 541623, 90019741563.0]

[356142, 365124, 413625, 61086262058.0]

[356142, 456213, 465213, 75639707982.0]

[356412, 415623, 435216, 68736115667.0]

[356421, 432651, 564132, 77096394077.0]

[361245, 421563, 461235, 72305936116.0]

[361245, 452613, 521463, 80199263303.0]

[361254, 462513, 613542, 83158148164.0]

[361254, 523461, 524361, 88830133177.0]

[361425, 526413, 615423, 94852868295.0]

[361425, 536142, 632415, 96781514366.0]

[361425, 536241, 631542, 96784793501.0]

[361524, 516243, 614325, 93185589278.0]

[361524, 523146, 613542, 94306452396.0]

[362145, 412563, 523146, 74382719720.0]

[362154, 416253, 534621, 75230234599.0]

[362154, 431562, 523461, 77389420170.0]

[362415, 623451, 625143, 108255794174.0]

[362451, 516432, 635421, 93579847366.0]

[362514, 416325, 426135, 68895482081.0]

[362514, 465213, 653214, 81986673302.0]

[362514, 612453, 652341, 109150392303.0]

[362541, 426315, 514263, 76312677732.0]

[362541, 514236, 531642, 88813590145.0]

[362541, 526431, 534261, 90325112009.0]

[362541, 631425, 641532, 110553693108.0]

[364125, 413526, 432165, 69391831124.0]

[364125, 425631, 532641, 77126616924.0]

[364125, 513246, 641325, 93364640287.0]

[364152, 523614, 645231, 95307676283.0]

[364215, 415362, 526431, 75314809249.0]

[364215, 651324, 652143, 113957896334.0]

[364251, 415236, 432165, 69598703862.0]

[364251, 415362, 541236, 75585813877.0]

[364512, 521463, 531246, 89964370580.0]

[364521, 465123, 615342, 84453650463.0]

[364521, 543261, 634251, 98805796321.0]

[365124, 524136, 635421, 95681349846.0]

[365124, 634521, 643152, 111732383903.0]

[365124, 642351, 645321, 112711994120.0]

[365214, 521364, 561243, 92495110391.0]

[365214, 541236, 615324, 98111605188.0]

[365214, 562134, 625134, 101599410808.0]

[365412, 423651, 564321, 77391577736.0]

[412356, 536124, 541632, 102641841312.0]

[412365, 412653, 451623, 77953319710.0]

[412365, 463215, 621435, 95506021827.0]

[412365, 521346, 562341, 102802529721.0]

[412536, 541236, 563124, 105503151178.0]

[412536, 541326, 564123, 105611407724.0]

[412563, 512436, 634251, 105430253139.0]

[412635, 425613, 546213, 86803499486.0]

[412653, 416325, 641352, 84213799697.0]

[413256, 423516, 635124, 86492286887.0]

[413256, 521436, 542163, 101123572806.0]

[413526, 431526, 536241, 87507344340.0]

[413526, 436521, 461253, 82233667662.0]

[413562, 456132, 543621, 91976621332.0]

[413562, 463152, 523146, 91597094832.0]

[413562, 512463, 643512, 105854906933.0]

[413562, 541632, 613425, 109811783585.0]

[413652, 514326, 536214, 99723346165.0]

[413652, 536241, 634521, 110019974598.0]

[413652, 562341, 615432, 113094750259.0]

[413652, 631254, 642135, 124500317832.0]

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Posted by Larry on 2023-10-28 14:07:11

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