Problems & Puzzles: Puzzles
Puzzle 249. From Rudolf to Rodolfo (magic squares and pandigital numbers)
In 1989 Rudolf Ondrejka (JMR, 21, Vol.1) asked:
Rodolfo Marcelo Kurchan, from Buenos Aires, Argentina, found (year?) the following answer to the Ondrejka's challenge:
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.
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
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!).
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.
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).
Here are their contributions in large.
C. Rivera wrote:
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:
And my PC is still working on...
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:
Jon Wharf wrote:
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
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
I think that this solution is not the smallest but now...I stop
the search ...