finding seven disjoint seven element subsets of {1,2, …, 49} with same sum

Question

I have a set containing numbers $1$ to $49$ $\{1,2,3, \cdots, 49\}$.

Now, I want to divide the set into $7$ subsets such that each subset should contain $7$ elements and sum of the elements of each subset should be $175$.

Is it possible to prove that such subsets exist?

Answer 1

Lookup Magic Squares.

which has the following:

22  47  16  41  10  35  4
5    23  48  17  42  11  29
30  6   24  49  18  36  12
13  31  7   25  43  19  37
38  14  32  1   26  44  20
21  39  8   33  2   27  45
46  15  40  9   34  3   28

Answer 2

Magic squares are fine, but here any single 7x7 Latin square works. An $n\times n$ Latin square is a square with the property that all the integers $1\ldots n$ appear exactly once on each row and column. There are several ways of constructing these. When $n=7$ we have for example the following $$ \begin{array}{ccccccc} 1&2&3&4&5&6&7\\ 7&1&2&3&4&5&6\\ 6&7&1&2&3&4&5\\ 5&6&7&1&2&3&4\\ 4&5&6&7&1&2&3\\ 3&4&5&6&7&1&2\\ 2&3&4&5&6&7&1\\ \end{array} $$

Given an $n\times n$ Latin square we can construct a solution to this problem (or its generalization of partitioning the integers $\{1,2,\ldots,n^2\}$ into $n$ equal sum groups of $n$ each) as follows.

Add $n(i-1)$ to all the entries on row $\#i$. After that sum of the entries on any column equals $\sum_{i=1}^ni+ n\left(\sum_{i=1}^{n}(i-1)\right)$, which is manifestly independent of the column. Furthermore, the integer $n(i-1)+j$, $1\le i,j\le n$, can only appear on row $\#i$, and does occur there exactly once (wherevere the entry $j$ of that row of the Latin square resides). The above 7x7 Latin square gives rise to the grouping gives rise to the group $$\{1,7+7=14,6+14=20,5+21=26,4+28=32,3+35=38,2+42=44\}$$ from the first column, $\{2,8,21,27,33,39,45\}$ from the second, $\{3,9,15,28,34,40,46\}$ from the third column, and so forth.

It is also possible to construct magic squares from Latin squares, but then you need a little bit richer structure. You need two so called mutually ortogonal Latin squares (=MOLS).