Prime nested magic squares
Honoring the Matrioskas craftworks art.
During these last two weeks I became busy looking for
what I will call "nested prime magic squares".
As its name suggests a nested magic square points to
two or more magic squares such that the internal(s) one is(are)
contained inside other(s) external(s) one, according to certain
valid rules that maintain the magicity property.
But this time we will proceed in a new direction to
avoid the so called "bordered magic squares". These, already
known, may be seen
In the bordered magic squares the magic squares are
of different order, in a very precise manner: if the inner magic
square is of order n, then the next outer magic square is of order
n+2, and so on...
In our nested magic squares the inner and the
outer magic squares remain of the same order. The only changes are
in the integers inside the cells and differ of course in the magic constant.
Our nested magic squares are
based in a very basic property of the magic squares:
all the integers in the cells of one magic square are multiplied
by the same constant k or/and if all
the integers in the cells are added the same constant
q (or a combination of these two
operations) then the resulting square remains
The special feature in our nested magic
squares is that in the outer magic square is
evident what are the integers that
remain from the inner magic square or if you prefer, is
evident what digits are going to be
peeled off from the outer magic square in order to get the inner
A very simple and trivial example is given:
magic square (n)
magic square (m)
According to the example, in a pair of nested magic
squares as the exemplified above, every integer in the inner matrix
is n and every integer in the outer matrix is m such that m=L&n&R
and the general rule is m=(L*10^LN+n)*10^LR+R, where L is the
integer to be added at the left of n, LN is the largest quantity of
digits of n, R is the integer to be appended at the right of n and
LR is the quantity of digits of R.
It may happen that both L&R are positive integers,
but it may happen that one of them is missed.
If no other conditions are imposed then the nested
magic squares described up to here are trivial.
One simple way to avoid triviality is to demand that all
the integers are of special form.
For example, what if all the integers must be prime
Now, the nested prime magic squares are far
from triviality. Nevertheless they exist.
Analyzing all the possible prime
magic squares with primes inside less than 1000, I obtained 34 nested solutions:
20 type LR, 10 type L and 4 type R.
Here are the smallest example for each type.
All of these three examples of nested prime magic
squares 3x3 are of the type "grade 2" which means that
involves 2 magic squares each.
Of course it may exist nested prime magic squares
grade k, for k>2.
Next I show three examples of nested prime magic
squares 3x3, grade 3.
Here we show only the very outer magic squares. As a
matter of fact, we show each of them in two ways: a) in the left
side, we use colours in order to indicate what digits must be peeled
off to obtain the inner magic squares b) in the right side, we use
parentheses for the same purpose than the colours.
Q. Please send one solution of
prime nested magic square 3x3,