Problems & Puzzles: Puzzles
Puzzle 252. Kurchan squares
Let's remember first what a magic square is.
For example, for a square 3x3 filled with the first 9 natural numbers (1 to 9), there is only one magic square
The key word for a magic square is the sum operation. But what are the multiplication values of the elements for the same magic square, for each row, column and (main) diagonal, equal or unequal? The answer is that in general the multiplication values are not equal to a constant.
Now let's calculate the difference between the maximal and the minimal of these eight products and we will get 75 = 120 -45.
Let's define K(n) for a square array nxn as the difference of the maximal and the minimal products for each row, column and (main) diagonal.
Is there a square array 3x3 such that K(3) is itself a minimal quantity, let's say K°(3), when filled with the first n2 natural numbers?
This exactly the question that was posed by Rodolfo Kurchan (1989) whose answer, given by himself, is K°(3)=72, and the corresponding 3x3 square is this one:
I will call this kind of squares - filled with the first n2 X-type of numbers and having a minimal K°(n) value - Kurchan multiplicative squares or shortly a Kurchan squares (*)
He solved also the same question filling the square with the first n2 prime numbers
K°(3) = 518
While Kurchan says in his email (10/1/04) that this last answer may be improved, I verified exhaustively his two answers and I can assure that he has gotten the minimal K°(3) solutions for both ways of filling the 3x3 square (natural numbers and prime numbers).
Here is the Kurchan question:
Q1. Find K°(n) for n=4-10 for both ways of filling the squares (the first n2 natural numbers and the first n2 prime numbers)
Now I want to add four (4) questions.
More interested in the method than in the results, and -of course- avoiding the exhaustive approaches...
Q2. ...do you devise a smart approach in order to get the K°(n) values and the corresponding squares?
I have obtained specific squares - filled with the first n2 natural numbers - such that K(4)=188 and K(5)=3680. I'm almost sure that my K(4) is K°(4) and then it can not be improved, but perhaps my K(5) is not yet the proper K°(5), so probably it can be improved.
Q3. Can you improve my K°(n) values for n=4 and 5, and/or get the specific squares associated?
Two more and last issues related to the Kurchan squares are the following ones:
Q4. Is there a Kurchan nxn square such that it is at the same time a magical nxn square, if the square is filled with the first consecutive n2 a) integers, b) primes, c) X-type numbers?
Q5. Is there a n value such that K°(n)=0, if the square is filled with the first consecutive n2 a) integers, b) primes, c) X-type numbers?
Contributions came from Luke Pebody, J. K. Andersen and Carlos Rivera.
Luke Pebody confirmed that the K°(4) obtained by C. Rivera is correct (the best possible).
Carlos Rivera improved his own solution for K(5) from 3680 to 2610 (can you find what is this improved arrangement?)
J. K. Andersen wrote, for the question 5:
Carlos Rivera improved (2/2/04) his own solution for K(5) from 2610 to 2052 (can you find what is this improved arrangement?)
Anurag Sahay wrote (May, 2005):
Anurag Sahay wrote (Set. 05):
On Set 26, 05, Luke Pebody reported:
k(5)=1744, This is the best solution... I have searched all ranges [m, m+1, ..., n] where m,n are products of five numbers in the range [1-25], n-m<1744 and m^5<25!, n^5>25! for possible squares, and there is no such range.
The square will be published later on Anurag's request.