"Special" 5x5 Magic Squares

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Poblem Description

Given the twenty five numbers. 1,1,2,2,2,2,2,3,3,3,3,3,3,4,5,5,5,5,6,6,6,6,6,7,9, arrange them in a 5x5 grid  in such a way that each horizontal row, vertical column and the two main diagonals all sum to 20.

Background & Techniques

Here's a puzzle sent to me by a viewer who says it is from a set of questions written by inventor and computer pioneer Clive Sinclair and published as "Mensa Steps" in a magazine, perhaps "Design Technology" in 1984.

I have no idea how this puzzle would be solved with pencil and paper.  I call it  "special"  because the standard Magic Square definition requires that consecutive integers from 1 to N2 be used to fill an N x N grid.   There is a vast amount of information about magic squares on the web, (11,500 Google hits for "5x5 Magic Squares"),  so the solution to this one may be out there somewhere but I did not find it.

My program solves the program in the same way that a very fast human might do it.  I divided the solution search into  3 phases.:

Phase 1

First we find all 5 number subsets of the given 25 which sum to 20.  Each row, column or diagonal must be one of these sets arranged in some order.  There are more than 50,000 was to choose 5 numbers from 25 but most of them do not sum to 20 and, of those that do, many are duplicates because of the repeated  values in the input set.  When we are done filtering  there are only 30 unique sets to choose from. Every row and every column (as well as the two main diagonals), must consist of one of these 30 sets arranged in some order. 

The "in some order" part though means that there can be up 120 arrangements (permutations) of the 5 numbers.  For the set {2,3,4,5,6}, the first number can be any of the 5, the second can be any of the other 4, etc. so there are 5x 4 x 3 x 2 x 1= 120 arrangements.  Repeated numbers wit in the set reduce the number of choices. sot there may only be 30 or  60permutations for some sets - still a lot of checking. though.     

Phase 2

Phase 2 applies the other constraint that all 25 of the given numbers must be used any set of 5 rows or columns chosen from the Phase 1 results.     Phase 2 assembles potential full grids by building 5 groups of the Phase 1 "columns" which contain the correct number of each digit value Two "1"s, five "2"s , six "3"s,  etc. Solutions must come from these sets, although each of these "columns" will likely have to be rearranged to find rows and diagonals that sum to 20 also. Of the 142,000 ways to choose 5 of the 30 Phase 1 results, there are about  4800  which contain  all of the 25 given numbers exactly once but only 110 of these are unique.. 

Phase 3

One of 12 solutions from candidate column set  11279 22556 23366 23456 33356

We're getting closer, but not there yet.  We can take any of the Phase 2 results and build 5x5 grid in which each group of 5 digits becomes a column which sum to 20, but it is unlikely that the sums in the other direction (the rows) will all be 20,    Phase 3 works on one of the Phase 2 results at a time.  forming 5 columns and then permuting the numbers contained in each column looking for rows which all sum to 20.  Since there are 5! (=120) ways to arrange each column, there could be as many as 1205 (25 billion)  possibilities to check.  In practice, there are many fewer than that because, again, many of the permutations will have only 30 or 60 unique permutations because of duplicate number values in the column.  The search space is also reduced by abandoning any solution if the row sum of the columns placed so far exceeds 20.  I have not evaluated all 110 possibilities, but in the first and last 10 groups, produced  from 0 to 56 solutions each with run times per group ranging from 1 to 23 seconds. The program does not rearrange the columns during the search, only the row positions within each column, so there are likely hundreds or thousands more solutions than this version finds.

Non-programmers are welcome to read on, but may want to jump to bottom of this page to download the executable program now.

Programmer's Notes:

The program uses the TComboset class to produce  the required combinations and permutations.   Recompiling will require a download of any version of our DFF Library zip file.  

To save time during  searching, Phase 1 attaches a string list with all of the unique permutations to each of the 30 potential columns and Phase 3 reads this list rather than re-computing them each time.   Phase 3 process uses  calls to recursive local function GetNextPermute to add   columns to the potential solution  being tested.  As each permutation of each potential column is added, the row sums of the columns added so far is kept and if the row sum total exceeds 20, the search for that branch is abandoned.  This was a late addition which reduced search times from minutes to seconds for each Phase 2 line being tested. 

 The code is well commented and time is short, so I'll leave this section at that for now.

Running/Exploring the Program 

bulletDownload source
bulletDownload  executable
bulletDownload source code library (DFFLibV15) (required to recompile the program) 

Suggestions for Further Explorations



Original:  March 10, 2012

Modified:  July 29, 2017

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