Magic squares with the same number of digits

There are magic, antimagic, multimagic or satanic, and even alphamagic squares. There are also squares within squares, squares made by prime numbers and square made up by pandigital numbers like that achieved by Rodolfo Kurchan
The latter is unique, being made by Pandigital numbers, if you count the number of occurrences of each digit, we see that obviously is the same for all, that is, around the square there are exactly sixteen zeros, sixteen ones, etc. .
Based on this square I started to look for magic squares of order three in which all digits appear the same number of times.
So I found these magic squares:

a) Magic square in that each digit appears twice
 
  109     32     87
    54     76     98
    65   120     43

b) Magic squares in which each digit appears exactly three times
   Also in this squares each column and row each digit appears only once

 1645     203       987
   287     945     1603
   903   1687       245

 1542       306       978
   378       942     1506
   906     1578       342

 1560       342       978
   378       960     1542
   942     1578      360

 1560       348       972
   372       960     1548
   948     1572       360

 1572      360       984
   384      972     1560
  960     1584       372
 
The last magic square that I've found is the only one such in each column and row are digits that appears twice, but is the only one that in one diagonal each digit appears only once.

 1329     276       996
   534     867     1200
  738    1458       405 


c) Magic squares in which each digit appears exactly four times

 22950         204     18156
   8976     13770     18564
   9384     27336       4590

 14399       2244     10285
   4862       8976     13090
   7667     15708       3553

Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!) Search: author:meller  Displaying 1-10 of 299 results found. page 1 2 3 4 5 6 7 8 9 10 ... 30      Sort: relevance | references | number A161140 Numbers which can be expressed as the product of numbers made of only twos. +0 7 2, 4, 8, 16, 22, 32, 44, 64, 88, 128, 176, 222, 256, 352, 444, 484, 512, 704, 888, 968, 1024, 1408, 1776, 1936, 2048, 2222, 2816, 3552, 3872, 4096, 4444, 4884, 5632, 7104, 7744, 8192, 8888, 9768, 10648, 11264, 14208, 15488, 16384, 17776 (list; graph; listen; history; internal format) OFFSET 1,1 COMMENTS 44 is in the list because 44 = 2 * 22, 484 is in the list because 484 = 22 * 22. CROSSREFS Cf. A084034, A161141, A161142, A161143, A161144, A161145, A161146, A161147. KEYWORD nonn,base AUTHOR Claudio L Meller (claudiomeller(AT)gmail.com), Jun 03 2009 A161141 Numbers which can be expressed as the product of numbers made of only threes. +0 7 3, 9, 27, 33, 81, 99, 243, 297, 333, 729, 891, 999, 1089, 2187, 2673, 2997, 3267, 3333, 6561, 8019, 8991, 9801, 9999, 10989, 19683, 24057, 26973, 29403, 29997, 32967, 33333, 35937, 59049, 72171, 80919, 88209, 89991, 98901, 99999, 107811 (list; graph; listen; history; internal format) OFFSET 1,1 COMMENTS 99 = 3 * 33; 1089 = 33 * 33; 999 = 3 * 333. CROSSREFS Cf. A084034, A161140, A161142, A161143, A161144, A161145, A161146, A161147. KEYWORD nonn,base AUTHOR Claudio L Meller (claudiomeller(AT)gmail.com), Jun 03 2009 EXTENSIONS Corrected and extended by Claudio L Meller (claudiomeller(AT)gmail.com), Jun 06 2009

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Almost Square, Magics

The other day I was trying to solve a magic square, by hand, when suddenly I was wrong and I had to cross out a cell of the square, keeping a drawing like the following:

Upon seeing this drawing, I forgot the original problem and began to think if it was possible to fill the remaining cells with non-repeated numbers in such a way that the sum of the rows, columns and diagonals give the same sum, that is, form an almost square that be magical

After trying and trying I came to a solution.

And as always happens, one wants more, then I changed the black square of position and returned to look for a solution.

Once found the solutions, I thought if I could find solutions for 3 x 3 squares.

Here are the solutions found:

For 3x3 :

We see that only the middle one has all  positive numbers

The magic sums of this almost square, magic are 9, 21 and 0 respectively.

For 4x4

In this case, all values are positive numbers.

and the magic sums are 56, 65 and 83 respectively

Some questions that came up:

a) For the 3x3 almost square, can you obtain almost squares with all  positive numbers for all the models?

b) What is the smallest possible magic sum (using only positive numbers), for each model of almost square, 3x3 and 4x4?

69720375229712477164533808935312303556800

By Cliff Pickower