The sum of every row, column and diagonals is 3.complete the magic square
5/4. ........ 11/12
......... 1. .......
........ ...... 3/4
Answers
Answer:
8+c....1+c....6+c..............
.............3+c....5+c....7+c..............
.............4+c....9+c....2+c..............A magic square remains magic, if you change each numbers by a constant c. You add c on the left. You can also subtract, multiply or divide. You define 4x4-, 5x5-... squares correspondingly.
The Magic 3x3 Square top
You have 1+2+3+4+5+6+7+8+9=45. In a magic square you have to add 3 numbers again and again. Therefore the average sum of three numbers is 45:3=15. The number 15 is called the magic number of the 3x3 square.
You can also achieve 15, if you add the middle number 5 three times.
You can reduce 15 in a sum of three summands eight times:15=1+5+9
15=1+6+815=2+4+9
15=2+5+8 15=2+6+7
15=3+4+815=3+5+7
15=4+5+6The odd numbers 1,3,7, and 9 occur twice in the reductions, the even numbers 2,4,6,8 three times and the number 5 once.
Therefore you have to place number 5 in the middle of the magic 3x3 square. The remaining odd numbers have to be in the middles of a side and the even numbers at the corners.
Under these circumstances there are eight possibilities building a square:
All the eight squares change into each other, if you reflect them at the axes of symmetry. You count symmetric squares only once. Therefore there is only one magic 3x3 square.
The Magic 4x4 Squaretop
The magic number is (1+2+...+15+16):4 = 34.
The computer found 86 reductions of 34 to a sum of four summands with the numbers 1 to 16.34=01+02+15+16
34=01+03+14+16
34=01+04+13+16
34=01+04+14+15
34=01+05+12+16
34=01+05+13+15
34=01+06+11+16
34=01+06+12+15
34=01+06+13+14
34=01+07+10+16
34=01+07+11+15
34=01+07+12+14
34=01+08+09+16
34=01+08+10+15
34=01+08+11+14
34=01+08+12+13
34=01+09+10+14
34=01+09+11+13 34=01+10+11+12
34=02+03+13+16
34=02+03+14+15
34=02+04+12+16
34=02+04+13+15
34=02+05+11+16
34=02+05+12+15
34=02+05+13+14
34=02+06+10+16
34=02+06+11+15
34=02+06+12+14
34=02+07+09+16
34=02+07+10+15
34=02+07+11+14
34=02+07+12+13
34=02+08+09+15
34=02+08+10+14
34=02+08+11+13 34=02+09+10+13
34=02+09+11+12
34=03+04+11+16
34=03+04+12+15
34=03+04+13+14
34=03+05+10+16
34=03+05+11+15
34=03+05+12+14
34=03+06+09+16
34=03+06+10+15
34=03+06+11+14
34=03+06+12+13
34=03+07+08+16
34=03+07+09+15
34=03+07+10+14
34=03+07+11+13
34=03+08+09+14
34=03+08+10+13 34=03+08+11+12
34=03+09+10+12
34=04+05+09+16
34=04+05+10+15
34=04+05+11+14
34=04+05+12+13
34=04+06+08+16
34=04+06+09+15
34=04+06+10+14
34=04+06+11+13
34=04+07+08+15
34=04+07+09+14
34=04+07+10+13
34=04+07+11+12
34=04+08+09+13
34=04+08+10+12
34=04+09+10+11
34=05+06+07+16 34
Answer:
The sum of every row, column and diagonals is 3.complete the magic square
5/4. ........ 11/12
......... 1. .......
........ ...... 3/4