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

***

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?

***

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          

***

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.......

***

 

 

---