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 Cutting corners. (Posted on 2014-08-14)

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 No Rating

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 re(2): what is wanted? | Comment 4 of 5 |
(In reply to re: what is wanted? by broll)

Yes indeed, area = sqrt(a^2 * b^2 + b^2 * c^2 + a^2 * c^2) / 2.

The second lines, added below the original lines, show the sum of the squares of the pairwise products of a,b,c, and then that divided by the square of the area. I knew the sum of the squares of the products was a fourth power of linear size, so I had to divide by the square of the area, and voila.

`  1  2  8      9.0000000000   11           324             4  2  4  6     14.0000000000   12           784             4  2  6  8     26.0000000000   16          2704             4  2  8 10     42.0000000000   20          7056             4  4  6 10     38.0000000000   20          5776             4  2  3 16     29.0000000000   21          3364             4  2  4 16     36.0000000000   22          5184             4  6  8  9     51.0000000000   23         10404             4  2 10 12     62.0000000000   24         15376             4  4  8 12     56.0000000000   24         12544             4  3  8 14     61.0000000000   25         14884             4  4  7 16     66.0000000000   27         17424             4  5  8 14     69.0000000000   27         19044             4  2 12 14     86.0000000000   28         29584             4  4  6 18     66.0000000000   28         17424             4  4 10 14     78.0000000000   28         24336             4  6  8 14     74.0000000000   28         21904             4  8 10 11     81.0000000000   29         26244             4`

and it works regardless of whether the open side is integral:

`  1  2  4      4.5825756950    7            84             4  1  2  6      6.7823299831    9           184             4  2  3  4      7.8102496759    9           244             4  1  2  8      9.0000000000   11           324             4  1  4  6     12.5299640861   11           628             4  2  3  6     11.2249721603   11           504             4  2  4  5     11.8743420870   11           564             4  2  4  6     14.0000000000   12           784             4  1  2 10     11.2249721603   13           504             4  1  4  8     16.6132477258   13          1104             4  2  3  8     14.7309198627   13           868             4  2  4  7     16.1554944214   13          1044             4  2  5  6     16.9115345253   13          1144             4  3  4  6     16.1554944214   13          1044             4  2  4  8     18.3303027798   14          1344             4  1  2 12     13.4536240471   15           724             4`

 Posted by Charlie on 2014-08-15 10:07:19

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