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A quite even partition (Posted on 2005-08-28) Difficulty: 3 of 5
Can you partition the numbers 1, 2, 3, ... nē in n separate subsets, each with n numbers, all subsets having the same sum?

  Submitted by Federico Kereki    
Rating: 4.3333 (3 votes)
Solution: (Hide)
Write a matrix like this; notice the diagonal: 1, n+1, 2n+1, ..., and the way the terms "grow" cyclically and to the right:
1        2        3        ...   n
2n       n+1      n+2      ...   2n-1
3n-1     3n       2n+1     ...   3n-2
...      ...      ...      ...   ...
(n-1)n+2 (n-1)n+3 (n-1)n+4 ...  (n-1)n+1
Now subtract 0 from the first row (!!), subtract n from the second row, 2n from the third, and so on, and (n-1)n from the n-th row. You get:
1        2        3        ...   n
n        1        2        ...   n-1
n-1      n        1        ...   n-2
...      ...      ...      ...   ...
2        3        4        ...   1
Each column has a permutation of the numbers 1, 2, ... n, so the sum is the same. Hence, the columns of the original matrix are a solution.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re: The Odd-Sided Squares PrincipleBruno2005-09-01 14:06:32
Another half solutionMcWorter2005-08-31 18:27:44
re: My solutionKen Haley2005-08-31 06:00:28
SolutionMy solutionKen Haley2005-08-31 05:57:47
re: The Odd-Sided Squares PrincipleFederico Kereki2005-08-31 03:21:40
re: The Odd-Sided Squares PrincipleMcWorter2005-08-30 16:21:31
The Odd-Sided Squares PrincipleBruno2005-08-30 11:49:37
re(2): Calendar MagicMcWorter2005-08-29 16:32:01
re: Calendar MagicBractals2005-08-29 16:23:22
re(2): Another (similar) way (To Mc)pcbouhid2005-08-29 11:56:05
re: Another (similar) way (To Mc)McWorter2005-08-29 03:23:28
Another (similar) way (To Mc)pcbouhid2005-08-29 03:06:53
re: Calendar MagicMcWorter2005-08-29 02:50:05
A simpler explanationGamer2005-08-29 02:01:09
SolutionCalendar MagicMcWorter2005-08-29 00:11:06
Some ThoughtsA process for creating the subsetsBob Smith2005-08-28 22:25:42
SolutionBractals2005-08-28 16:39:14
Some ThoughtsHalf Solutionowl2005-08-28 14:54:58
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