Find the Master Magic Squares of 9x9 Magic Squares using Numbers from 1-81
This is a CASCADED VERSION with LOOSE ONION PEELS DESIGN in that : The Middle Core 3x3 Square is a Magic Square using 9 numbers from 1-81
This is enveloped by a 5x5 Magic Square using 16 more numbers in addition to the 9 numbers already used in the core 3x3 MS thus using a total of 25 numbers out of 1-81
This 5x5 MS is further enveloped by a 7x7 Magic Square adding another 24 numbers thus using a total of 49 numbers out of 1-81
Finally this 7x7 MS is enveloped by a 9x9 Overall Magic Square using all numbers 1-81
To illustrate please see the figure :
This illustrates how each colored Core forms a 3x3 5x5 7x7 & 9x9 Magic Square respectively !
It can also be seen that each of these Magic Squares as well as these Sleeves can be rotated & still have a Magic Square & therefore there are many Solutions Possible
To Illustrate further I give a Sample 9x9 1-81 Magic Square and analyze for illustration how it will add up in the smaller Squares within
We can see here a 9x9 Sample Magic Square and to help add the Columns Rows & Diagonals I have indicated their Totals too.
We can see that the Overall MS adds to the Magic Sum of 369 correctly where us the Component 3x3 5x5 & 7x7 Squares are not Magic Squares
The Solution to the Puzzle requires all of these Components to be Magic Squares with Most likely Magic Sums of 123, 205,287 & 369 respectively ! There could be many Solutions
Puzzle 1 : Find a Solution with 1-81 numbers so arranged that we have 3x3 5x5 7x7 & 9x9 Magic Squares cascaded like illustrated
Puzzle 2 : Find a Solution with 1-81 numbers so arranged that we have 3x3 5x5 7x7 & 9x9 Magic Squares cascaded like illustrated with each Magic Square having Sequential numbers. Like 3x3 MS with numbers 1-9 or 37 - 45 etc followed by 5x5 MS with 1-25 or 29-53 and so on.
Like in the above sample 9x9 MS has sequential numbers from 1-81
Similarly the component 3x3 5x5 & 7x7 each must have a sequential Block of numbers out of 1-81 3x3 MS need not start with 1 [perhaps can not]