Problems & Puzzles: Puzzles

 Puzzle 411. Magic squares M & p(M)  Now a very charming puzzle. Giovanni Resta has sent to me a magic square that he claims to be the smallest 3x3 magic square M such that if you change every entry k in M by p(k) you get a new magic square p(M), where p(k) is the k-th prime. Q1. Can you compute the first five smallest magic squares M having this property? Note that k is not asked to be necessarily a prime. Q2. Find nxn magic square having the same property, for n=4, 5 & 6, not necessarily minimal. Q3. Find the smallest 3x3 magic square M such that p(M) & p(p(M)) are magic too. By my side I will add just a fourth and perhaps simpler question than Q2 & Q3: Q4. Find the smallest 3x3 magic square M such that p(M) is magic but every entry k in M is prime.

Andreas Thomas Höglund found (Aug. 07) four solutions to Q2, for n=4. Here is his minimal solution with Magic sums 58 & 192 for M & P(M).

 4 15 29 10 7 47 109 29 16 18 5 19 53 61 11 67 17 22 13 6 59 79 41 13 21 3 11 23 73 5 31 83

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J. C. Rosa found (Aug, 31, 07) the same minimal solution already found by Resta, asked in Q1. Now is time to show it:

Magic square ( M )

26199       25436      25819
25438       25818      26198          magic sum=77454
25817       26200      25437

Magic square p(M)

302573        292667        297629
292679        297623        302567     magic sum=892869
297617        302579        292673

How far is the second smallest?

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J. C. Rosa added (Sept 6, 07)

Here is the second smallest solution of Q1:

Magic square ( M )

30579       29536      30059
29538       30058      30578          magic sum=90174
25817       26200      2543

Magic square p(M)

357683        344417        351059
344429        351053        357677     magic sum=1053159
351047        357689        344423

On Set. 20, 07, he wrote again:

Here is the third smallest solution :

Magic square ( M )

34063       30099      32108
30135       32090      34045          magic sum=96270
32072       34081      30117

Magic square p(M)

402797        351551        377543
352043        377297        402551     magic sum=1131891
377051        403043        351797

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Finally on Oct 30 he wrote:

Here are the fourth and the fifth smallest solutions of the
question 1 :

4°)                 Magic square ( M )

49765       42208      45988
42210       45987      49764          magic sum=137961
45986       49766      42209

Magic square p(M)

608831        508919        558893
508943        558881        608819     magic sum=1676643
558869        608843        508931

5°)                     Magic square ( M )

47482       44839      46162
44841       46161      47481          magic sum=138483
46160       47483      44840

Magic square p(M)

578603        543497        561059
543509        561053        578597     magic sum=1683159
561047        578609        543503

Now  the question is completely solved and I can try to solve
the question 4.......

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