Q4. Complete the magic square:
The magic number is 12.6
Answers
I found 7040 solutions.
You might think that number is very big, but the number of ways you can arrange a set of 16 numbers in 16 positions is much much bigger… this is 16! = 16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1 … a huge number that is more than 2*10^13, that is, more than 20 trillion permutations. Of course, most of the permutations don’t form a magic square, only 7040 of them do form it, which is only 1 in every 2971987200, that is, roughly 1 in every 3 billion.
Since a 4x4 square has 4*4 = 16 numbers, usually the magic squares are searched in the 0 to 15 range, that is, with the lowest non-negative values.
Note that 0+1+2+3+4+…+13+14+15 = (0+15) + (1+14) + … +(7+8) = 15*8 =120
If all the 16 numbers add up to 120, then in a magic square every row or column will have a sum of 30 = 120/4 (there are 4 rows and 4 columns).
This is the complete list of 7040 solutions (with numbers from 0 to 15):
Answer:
Solution 2:
(i) Row-wise sum:
6 + 7 + 2 = 15
1 + 5 + 9 = 15
8 + 3 + 4 = 15
Column-wise sum:
6 + 1 + 8 = 15
7 + 5 + 3 = 15
2 + 9 + 4 = 15
Diagonal-wise sum:
6 + 5 + 4 = 15
2 + 5 + 8 = 15
(ii) Row-wise sum:
4 + 9 + 8 = 21
11 + 7 + 3 = 21
6 + 5 + 10 = 21
Column-wise sum:
4 + 11 + 6 = 21
9 + 7 + 5 = 21
8 + 3 + 10 = 21
Diagonal-wise sum:
4 + 7 + 10 = 21
8 + 7 + 6 = 21
(iii) Row-wise sum:
16 + 2 + 12 = 30
6 + 10 + 14 = 30
8 + 18 + 4 = 30
Column-wise sum:
16 + 6 + 8 = 30
2 + 10 + 18 = 30
12 + 14 + 4 = 30
Diagonal-wise sum:
16 + 10 + 4 = 30
12 + 10 + 8 = 30