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 n^{2
}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 GonzalezMorris
told him that this was now the smallest (just for the 4x4 case, as you will
learn in short).
German GonzalezMorris 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 10digit pandigital numbers which
are also 10digit 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 ...
***
