An Inverted Magic Square

Using the digits 0, 1, 6, 8 and 9, fill a 5x5 grid with 2-digit numbers (leading zeros are allowed, and necessary), such that:

Source: Based on a problem from The Little Giant Encyclopedia of Puzzles.


Solutions were received from John Hewson, Robert Kelnhofer, Jayavel Sounderpandian, and Ken Suprin. John Hewson's solution is below. He points out that a solution will work as long as each of the five digits is found once in each row, column, and diagonal, in both the units and tens digits. A simple way to do this is to place copies of each digit at knight-move locations from each other:
A B C D E
C D E A B
E A B C D
B C D E A
D E A B C
Replacing ABCDE with 0,1,6,8,9 will place the units digits properly. Using the mirror image of this pattern and again replacing the letters with the digits (in any order) will place the tens digits and result in creating all 25 unique numbers. Then each row, column and diagonal will sum to (10+1)*(0+1+6+8+9) = 264.

Ken Suprin sent these examples:

09 90 81 66 18 
86 68 19 00 91 
10 01 96 88 69 
98 89 60 11 06 
61 16 08 99 80 
       
 
08 99 80 61 16 
60 11 06 98 89 
96 88 69 10 01 
19 00 91 86 68 
81 66 18 09 90 

John Hewson's solution:
Result:
00  91  86  68  19    
66  18  09  90  81
99  80  61  16  08
11  06  98  89  60
88  69  10  01  96
is a 5x5 magic square fulfilling the requirements.

Method:
There are just 25 two digit numbers which use the digits 0, 1, 6, 8, and 
9.   They are:
00  01  06  08  09
10  11  16  18  19
60  61  66  68  69
80  81  86  88  89
90  91  96  98  99
The sum of these two digit numbers is 1320 and thus the magic square using
each once only has magic constant 1320/5 = 264.   When this array of
numbers is inverted the same set of numbers (in a different order) results
and thus the magic constant of an inverted magic square is likewise 264.  
The 5 numbers 0, 1, 6, 8 and 9 sum to 24 and 10 x 24 + 24 =264, so that a
sequence of five of the two digit numbers will sum to 264 and thus be
suitable for a row, column or diagonal of a magic square, of the required
type, if each of the five digits occurs once in the tens position and once
in the units position of the five numbers.   Such five number sequences can
be read from the main and broken diagonals of the above array to form the
rows:

(diagonals from SW to NE)
00  91  86  68  19
10  01  96  88  69
60  11  06  98  89
80  61  16  08  99
90  81  66  18  09

(diagonals from NW to SE)  
90  01  16  68  89
80  91  06  18  69
60  81  96  08  19
10  61  86  98  09
00  11  66  88  99
Each of the 25 2-digit numbers occurs once in the SW to NE array and once
in the NW to SE array and the rows of the first of these arrays and the
rows of the second can be used as the rows and columns respectively of a
magic square provided the common numbers are aligned and the other numbers
in each row are permuted cyclically.   This results in an array which is
magic apart from diagonals:
00  91  86  68  19
11  06  98  89  60
66  18  09  90  81
88  69  10  01  96
99  80  61  16  08

Finally to ensure that the diagonals sum to 264 take the set of five
numbers:  00 61 18 89 96 (selected by a sequence of Knight's moves in the
first array of this method section), and rearrange the rows of the above
square so that those numbers, in some order, form the NW to SE main
diagonal:
00  91  86  68  19
66  18  09  90  81
99  80  61  16  08
11  06  98  89  60
88  69  10  01  96
The SW to NE diagonal then also has the required sum.

The inverse of this magic square:
96  10  01  69  88
09  68  86  90  11
80  91  19  08  66
18  06  60  81  99
61  89  98  16  00

is also magic, as required, with magic constant 264.

Mail to Ken