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In the square below, change the position of six numbers, one per horizontal row, vertical column and long diagonal line of six smaller squares, in such a way so the numbers in each row, column or diagonal line total exactly 246. Any number may appear more than once in each row, column or line.

Attribution: E Saunders, D. Turing Expert Number Puzzles

39 13 24 25 68 39
38 44 66 15 41 45
41 74 41 23 41 46
33 49 41 59 20 59
36 43 21 58 39 45
63 26 49 28 57 27
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    $\begingroup$ Is this an original puzzle? If not, it requires proper attribution. $\endgroup$
    – Herb
    Commented Sep 14, 2021 at 18:49
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    $\begingroup$ This exercise seems to have quite a lot of additions and not all that much actual puzzle solving to it. $\endgroup$
    – Bass
    Commented Sep 15, 2021 at 19:54

1 Answer 1

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This puzzle is easier than it looks.

First you have to realise that

In each column and each row, only one cell will be changed. So each changing cell will change its row sum and its column sum by the same amount, and those sums are not changed by any other cells. Therefore the row sums must match the column sums in some order, and their intersections are the cells that have to be changed.

We then simply look at the row and column sums:

I have marked the cells that are in a row and column with an equal sum with hash signs.

                                250
       39  13  24 #25# 68  39 | 208
       38 #44# 66  15  41  45 | 249
       41  74  41  23 #41# 46 | 266
       33  49  41  59  20 #59#| 261
       36  43 #21# 58  39  45 | 242
      #63# 26  49  28  57  27 | 250
      ------------------------
      250 249 242 208 266 261   249

Now that we know which cells have to change, it is simple matter to change them to make the sums equal to 246. These changes indeed happen to be a permutation of those six numbers.

                                246
       39  13  24  63  68  39 | 246
       38  41  66  15  41  45 | 246
       41  74  41  23  21  46 | 246
       33  49  41  59  20  44 | 246
       36  43  25  58  39  45 | 246
       59  26  49  28  57  27 | 246
      ------------------------
      246 246 246 246 246 246   246

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