Problems & Puzzles: Puzzles

Puzzle 249.  From Rudolf  to Rodolfo (magic squares and pandigital numbers)

In 1989 Rudolf Ondrejka (JMR, 21, Vol.1) asked:

what is the magic square with the smallest magic sum using only pandigital numbers?

Rodolfo Marcelo Kurchan, from Buenos Aires, Argentina, found (year?) the following answer to the Ondrejka's challenge:

 1037956284 1036947285 1027856394 1026847395 1026857394 1027846395 1036957284 1037946285 1036847295 1037856294 1026947385 1027956384 1027946385 1026957384 1037846295 1036857294

Pandigital magic sum = 4129607358

Kurchan says that he found his solution without using computer.

I found this magic square at the page 237 of the C. A. Pickover's 'Wonders of numbers'. But you can see it also in one of the Kurchan's pages at the web.

Pickover writes:

"He [Kurchan] believes that this is the smallest nontrivial magic square having n2 distinct pandigital (*) integers and having the smallest pandigital magic sum".

I think that this is not so; probably the above shown magic square is the smallest magic 4x4 of that type, but it must exist some 3x3 solution.

As a matter of fact I have gotten without too much pain ( because I used my PC and codes ;-) a 3x3 solution of the same type just disregarding the pandigital magic sum condition:

1023856974 1032857469 1028356479
1032856479 1028356974 1023857469
1028357469 1023856479 1032856974

Magic sum = 3085070922 (non pandigital)

I suspect that near to this one it should exist another solution with a pandigital magic sum (but I might be wrong!)

Question 1. Find the smallest 3x3 magic square as the Kurchan' s 4x4 one (if it exist!).

Question 2. Find a 3x3 magic square using only primes each having all the ten digits at least once and with the magic sum of the same type (but composite, of course!).

_________
(*)pandigital means here that all ten digits are used and 0 is not a leading digit.

Solution:

For the Question 1 contributions came from Rodolfo Marcelo Kurchan, C. Rivera, J. C. Rosa and Jon Wharf.

Only C. Rivera and J. C. Rosa discovered technically at the same time and independently, the asked (minimal) solution to Question 1.

Nobody has sent specific solutions to Question 2.

A Happy and unexpected note! Rodolfo Marcelo Kurchan was contacted by email and sent an improved solution by himself obtained recently, for the 4x4 pandigital magic square with pandigital magic sum.

 1034728695 1035628794 1024739685 1025639784 1024639785 1025739684 1034628795 1035728694 1035629784 1034729685 1025638794 1024738695 1025738694 1024638795 1035729684 1034629785

Pandigital magic sum = 4120736958. He says that German Gonzalez-Morris told him that this was now the smallest (just for the 4x4 case, as you will learn in short).

German Gonzalez-Morris added (May 2006) that he made a computer program and found an smaller pandigital sum (4120967358) then Rodolfo (by hand) found the smallest sum (4120736958), finally German found (and prove by exhaustive search) all smallest sums beginning from: 4120736958, 4120953678, 4120967358, 4127360958, 4129536078, ...

Here are their contributions in large.

***

C. Rivera wrote:

As a matter of fact, as I suspected there is one smaller (than the Kurchan's one) pandigital magic sum solution in a magic 3x3 square:

1057834962 1084263579 1063549278
1074263589 1068549273 1062834957
1073549268 1052834967 1079263584
Pandigital Magical sum = 3205647819

I got it this Sunday morning (4/1/04). It was pretty close enough the one reported before when I posed this puzzle the Saturday morning. So, my PC just worked 24 hours more and bingo!. By the method employed (exhaustive and upward) this must be the minimal solution.

Other solutions after the minimal one and still less that the Kurchan one (shown in increasing pandigital magic sum) are:

1089362475 1320589746 1204968537
1320579648 1204973586 1089367524
1204978635 1089357426 1320584697
Pandigital Magical sum = 3614920758

1084793625 1327405896 1205349687
1326405798 1205849736 1085293674
1206349785 1084293576 1326905847
Pandigital Magical sum = 3617549208

1085793462 1328405679 1206349578
1327405689 1206849573 1086293457
1207349568 1085293467 1327905684
Pandigital Magical sum = 3620548719

1045793862 1368405279 1206349578
1367405289 1206849573 1046293857
1207349568 1045293867 1367905284
Pandigital Magical sum = 3620548719

1045798362 1368420579 1206359478
1367420589 1206859473 1046298357
1207359468 1045298367 1367920584
Pandigital Magical sum = 3620578419

And my PC is still working on...

Notes:

a) Please observe my 4th and 5th solution: they share the same pandigital magical sum!

b) But the problem posed by Ondrejka is a kind of old (15 years!), so I also suspect that someone else should have gotten the minimal solution before and of course that I'll be glad to publish the name of the first discoverer properly referenced...

***

J. C. Rosa wrote:

Today (Wednesday 7/1/04) is a magic day. I have found the smallest 3x3 magic square with the smallest magic sum using only pandigital numbers . Here it is :

1079263584 1052834967 1073549268
1062834957 1068549273 1074263589
1063549278 1084263579 1057834962
Magic sum =3205647819

Now , I'm looking for the largest....

***

Jon Wharf wrote:

After thinking about active groups of digits in a magic square and playing with bits of paper for ages, I generated the 5820 10-digit pandigital numbers which are also 10-digit pandigital when multiplied by 3.

So pretty quickly after that I found one solution:

 1720945863 1270946853 2170946358 2170946853 1720946358 1270945863 1270946358 2170945863 1720946853

with pandigital magic constant 5162839074.

Minimum? no, but at least we're started....

Next solution uncovered was:

 1283604759 1238704659 1328654709 1328704659 1283654709 1238604759 1238654709 1328604759 1283704659

with pandigital magic constant 3850964127. This was the smallest I found. It has the definite virtue of a smaller magic constant than Rodolfo's.

***

J. C. Rosa wrote (March 23, 2005):

I have found (at last !) a solution to the question 2,

but I think that this solution maybe  is not the smallest ...

10887852687493    10245252478639    10575552896347
10257252896347    10569552687493    10881852478639
10563552478639    10893852896347    10251252687493

magic sum=31708658062479

***

Later, on May 5, 2005 he wrote too:

About the question 2 of the puzzle 249 I have found
several solutions smaller than the one already published.
Here is my best solution ( with 9 prime pandigital numbers
of 12 digits each ):

914052876349      106438267459      510267485239
106467485239      510252876349      914038267459
510238267459      914067485239      106452876349

magic sum=1530758629047

I think that this solution is not the smallest but now...I stop
the search ...

***

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