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Cutting corners. (Posted on 2014-08-14) Difficulty: 3 of 5

I cut off one corner of an empty box to form an open 'pocket', so that each of the length a, breadth b, and depth c, of the initial corner was a different whole number of centimetres, and the area of each face of the 'pocket' was also an integer.

What is the area of the open 4th side of the 'pocket', in terms of a,b, and c?

See The Solution Submitted by broll    
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Some Thoughts what is wanted? | Comment 1 of 5
It's hard to say what's wanted here.

I wrote a program that finds the area of the open face based on using the pythagorean theorem to find the lengths of the sides of the open triangular face, and then Heron's formula for finding the area of the triangle.

The only restriction given is basically that at least two out of a, b and c must be even.

c' = sqrt(a^2 + b^2)
b' = sqrt(a^2 + c^2)
a' = sqrt(c^2 + b^2)

s = (a' + b' + c')/2

area = sqrt(s * (s - a') * (s - b') * (s - c'))

Is what's wanted more than just appropriately substituting a, b and c within the last formula?

Or is there a simpler formula looked for, perhaps based on those situations were the 4th, open, side also has integral area, shown among the below examples.

Some examples:

  a  b  c    area          a+b+c
  1  2  4    4.5825756950    7
  1  2  6    6.7823299831    9
  1  4  4    8.4852813742    9
  2  3  4    7.8102496759    9
  2  4  4    9.7979589711   10
  1  2  8    9.0000000000   11
  1  4  6   12.5299640861   11
  2  3  6   11.2249721603   11
  2  4  5   11.8743420870   11
  3  4  4   11.6619037897   11
  2  4  6   14.0000000000   12
  1  2 10   11.2249721603   13
  1  4  8   16.6132477258   13
  1  6  6   18.4932420089   13
  2  3  8   14.7309198627   13
  2  4  7   16.1554944214   13
  2  5  6   16.9115345253   13
  3  4  6   16.1554944214   13
  2  4  8   18.3303027798   14
  2  6  6   19.8997487421   14
  1  2 12   13.4536240471   15
  1  4 10   20.7123151772   15
  1  6  8   24.5153013443   15
  2  3 10   18.2756668825   15
  2  4  9   20.5182845287   15
  2  5  8   22.1133443875   15
  2  6  7   22.9346898824   15
  3  4  8   20.8806130178   15
  3  6  6   22.0454076850   15
  4  5  6   21.6564078277   15
  2  4 10   22.7156333832   16
  2  6  8   26.0000000000   16
  4  6  6   24.7386337537   16
  1  2 14   15.6843871414   17
  1  4 12   24.8193472920   17
  1  6 10   30.5614135799   17
  1  8  8   32.4961536185   17
  2  3 12   21.8403296678   17
  2  4 11   24.9198715888   17
  2  5 10   27.3861278753   17
  2  6  9   29.0860791445   17
  2  7  8   29.9499582637   17
  3  4 10   25.7099202644   17
  3  6  8   28.3019433962   17
  4  5  8   27.4954541697   17
  4  6  7   27.9463772250   17
  5  6  6   27.8208554865   17
  2  4 12   27.1293199325   18
  2  6 10   32.1869538789   18
  2  8  8   33.9411254970   18
  4  6  8   31.2409987036   18
  1  2 16   17.9164728672   19
  1  4 14   28.9309522830   19
  1  6 12   36.6196668472   19
  1  8 10   40.5092582011   19
  2  3 14   25.4165300543   19
  2  4 13   29.3428015022   19
  2  5 12   32.6955654485   19
  2  6 11   35.2987251894   19
  2  7 10   37.0675059857   19
  2  8  9   37.9605057922   19
  3  4 12   30.5941170816   19
  3  6 10   34.7275107084   19
  3  8  8   36.2215405525   19
  4  5 10   33.5410196625   19
  4  6  9   34.5976877840   19
  4  7  8   35.1567916625   19
  5  6  8   34.6554469023   19
  2  4 14   31.5594676761   20
  2  6 12   38.4187454246   20
  2  8 10   42.0000000000   20
  4  6 10   38.0000000000   20
  4  8  8   39.1918358845   20
  1  2 18   20.1494416796   21
  1  4 16   33.0454232837   21
  1  6 14   42.6848919408   21
  1  8 12   48.5386443980   21
  1 10 10   50.4975246918   21
  2  3 16   29.0000000000   21
  2  4 15   33.7786915081   21
  2  5 14   38.0263066837   21
  2  6 13   41.5451561557   21
  2  7 12   44.2379927212   21
  2  8 11   46.0543157587   21
  2  9 10   46.9680742633   21
  3  4 14   35.5105618091   21
  3  6 12   41.2431812546   21
  3  8 10   44.3734154647   21
  4  5 12   39.6988664826   21
  4  6 11   41.4366987102   21
  4  7 10   42.6731765867   21
  4  8  9   43.3128156554   21
  5  6 10   41.8330013267   21
  5  8  8   42.7083130081   21
  6  7  8   42.4381903478   21
  2  4 16   36.0000000000   22
  2  6 14   44.6766158074   22
  2  8 12   50.1198563446   22
  2 10 10   51.9615242271   22
  4  6 12   44.8998886413   22
  4  8 10   47.4973683482   22
  6  8  8   46.6476151588   22
  1  2 20   22.3830292856   23
  1  4 18   37.1618083521   23
  1  6 16   48.7544869730   23
  1  8 14   56.5773806393   23
  1 10 12   60.5061980296   23
  2  3 18   32.5883414736   23
  2  4 17   38.2230297072   23
  2  5 16   43.3704968844   23
  2  6 15   47.8121323515   23
  2  7 14   51.4392845984   23
  2  8 13   54.1940956194   23
  2  9 12   56.0446250768   23
  2 10 11   56.9736781330   23
  3  4 16   40.4474968323   23
  3  6 14   47.8121323515   23
  3  8 12   52.6497863244   23
  3 10 10   54.3139024560   23
  4  5 14   45.9238500128   23
  4  6 13   48.3838816136   23
  4  7 12   50.3587132481   23
  4  8 11   51.7300686255   23
  4  9 10   52.4309069157   23
  5  6 12   49.2036584006   23
  5  8 10   51.2347538298   23
  6  7 10   50.6557005677   23
  6  8  9   51.0000000000   23
  7  8  8   50.9116882454   23
  2  4 18   40.4474968323   24
  2  6 16   50.9509568114   24
  2  8 14   58.2752091373   24
  2 10 12   62.0000000000   24
  4  6 14   51.8844870843   24
  4  8 12   56.0000000000   24
  4 10 10   57.4456264654   24
  6  8 10   55.4616984954   24
  1  2 22   24.6170672502   25
  1  4 20   41.2795348811   25
  1  6 18   54.8270006475   25
  1  8 16   64.6219776856   25
  1 10 14   70.5265907300   25
  1 12 12   72.4982758416   25
  2  3 20   36.1801050303   25
  2  4 19   42.6731765867   25
  2  5 18   48.7237108603   25
  2  6 17   54.0925133452   25
  2  7 16   58.6600375042   25
  2  8 15   62.3618473107   25
  2  9 14   65.1613382306   25
  2 10 13   67.0373030484   25
  2 11 12   67.9779375974   25
  3  4 18   45.3982378513   25
  3  6 16   54.4150714417   25
  3  8 14   61.0000000000   25
  3 10 12   64.4127316608   25
  4  5 16   52.1919534028   25
  4  6 15   55.3985559379   25
  4  7 14   58.1463670404   25
  4  8 13   60.2992537267   25
  4  9 12   61.7737808459   25
  4 10 11   62.5219961294   25
  5  6 14   56.6921511322   25
  5  8 12   60.0333240792   25
  5 10 10   61.2372435696   25
  6  7 12   59.1692487699   25
  6  8 11   60.0083327547   25
  6  9 10   60.4483250388   25
  7  8 10   60.0749531835   25
  2  4 20   44.8998886413   26
  2  6 18   57.2363520850   26
  2  8 16   66.4529909033   26
  2 10 14   72.0832851638   26
  2 12 12   73.9729680356   26
  4  6 16   58.9236794506   26
  4  8 14   64.6219776856   26
  4 10 12   67.6461381012   26
  6  8 12   64.6219776856   26
  6 10 10   65.5743852430   26
  1  2 24   26.8514431642   27
  1  4 22   45.3982378513   27
  1  6 20   60.9015599143   27
  1  8 18   72.6704891961   27
  1 10 16   80.5543294926   27
  1 12 14   84.5044377533   27
  2  3 22   39.7743636027   27
  2  4 21   47.1274866718   27
  2  5 20   54.0832691320   27
  2  6 19   60.3821165578   27
  2  7 18   65.8938540381   27
  2  8 17   70.5478560978   27
  2  9 16   74.3034319530   27
  2 10 15   77.1362431027   27
  2 11 14   79.0316392339   27
  2 12 13   79.9812478022   27
  3  4 20   50.3587132481   27
  3  6 18   61.0409698481   27
  3  8 16   69.3974062916   27
  3 10 14   74.6056298144   27
  3 12 12   76.3675323681   27
  4  5 18   58.4893152636   27
  4  6 17   62.4579858785   27
  4  7 16   66.0000000000   27
  4  8 15   68.9637585983   27
  4  9 14   71.2530701093   27
  4 10 13   72.8079665971   27
  4 11 12   73.5934779719   27
  5  6 16   64.2572953057   27
  5  8 14   69.0000000000   27
  5 10 12   71.5891053164   27
  6  7 14   67.8675180038   27
  6  8 13   69.2892488053   27
  6  9 12   70.2922470832   27
  6 10 11   70.8096038684   27
  7  8 12   69.6562990691   27
  7 10 10   70.3562363974   27
  8  9 10   70.1498396292   27
  2  4 22   49.3558507170   28
  2  6 20   63.5295206971   28
  2  8 18   74.6458304261   28
  2 10 16   82.1948903521   28
  2 12 14   86.0000000000   28
  4  6 18   66.0000000000   28
  4  8 16   73.3212111193   28
  4 10 14   78.0000000000   28
  4 12 12   79.5989949685   28
  6  8 14   74.0000000000   28
  6 10 12   76.1314652427   28
  8 10 10   75.4983443527   28
  1  2 26   29.0860791445   29
  1  4 24   49.5176736126   29
  1  6 22   66.9776081986   29
  1  8 20   80.7217442824   29
  1 10 18   90.5869747811   29
  1 12 16   96.5194280961   29
  1 14 14   98.4987309563   29
  2  3 24   43.3704968844   29
  2  4 23   51.5848815061   29
  2  5 22   59.4474557908   29
  2  6 21   66.6783323127   29
  2  7 20   73.1368580129   29
  2  8 19   78.7464284904   29
  2  9 18   83.4625664595   29
  2 10 17   87.2582374335   29
  2 11 16   90.1165911472   29
  2 12 15   92.0271699011   29
  2 13 14   92.9838695689   29
  3  4 22   55.3263047745   29
  3  6 20   67.6830850361   29
  3  8 18   77.8267306264   29
  3 10 16   84.8587060943   29
  3 12 14   88.4364178379   29
  4  5 20   64.8074069841   29
  4  6 19   69.5485441976   29
  4  7 18   73.8985791474   29
  4  8 17   77.6916983982   29
  4  9 16   80.8207894047   29
  4 10 15   83.2165848855   29
  4 11 14   84.8351342310   29
  4 12 13   85.6504524215   29
  5  6 18   71.8748913043   29
  5  8 16   78.0768851838   29
  5 10 14   82.1583836258   29
  5 12 12   83.5703296631   29
  6  7 16   76.6876782802   29
  6  8 15   78.7464284904   29
  6  9 14   80.3865660418   29
  6 10 13   81.5230028888   29
  6 11 12   82.1035931004   29
  7  8 14   79.5047168412   29
  7 10 12   81.1726554943   29
  8  9 12   80.7217442824   29
  8 10 11   81.0000000000   29
  9 10 10   80.9320702812   29
  2  4 24   53.8144961883   30
  2  6 22   69.8283610004   30
  2  8 20   82.8492607088   30
  2 10 18   92.3255111007   30
  2 12 16   98.0612053771   30
  2 14 14   99.9799979996   30
  4  6 20   73.1026675300   30
  4  8 18   82.0731381147   30
  4 10 16   88.4533775500   30
  4 12 14   91.7387595294   30
  6  8 16   83.5224520713   30
  6 10 14   86.9712596206   30
  6 12 12   88.1816307402   30
  8 10 12   86.6256313108   30

  Posted by Charlie on 2014-08-14 16:34:12
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