All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Shapes
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    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution 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
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (3)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information