Question - help!
[Originally posted in Discussions]
I actually did it another way. In this case, I knew I can use each number from 1 - 7 only ones.
Out of those numbers, there are 3 even numbers and 4 odd numbers. 9 which is an odd numbers is in the middle and 8 which is an even numbers is on the edge. And there are exactly 3 spaces on the edge and total 4 spaces in the center of row and colum including the center.
It proves that only even numbers will go on the edges and odd will go on the centers. The biggest even number in this case after 8 is 6, so it is more likely that it might sit on the bottom right where 8 is. Then I could try out from 2 or 4 and see whether its sum is the same like others.
I got the answer right, though.
I understand it will take time in this way. But for the second half of the method of finding the middle number, what if the magic square is not a square. What if I was a rectangle or any kind of strange shape? How would then I draw the lines and find out how many of them shares that center?
Can I also use the method I used while thinking?
I commend you for thinking deeply about this problem! Your approach to consider where the odd/even numbers should go is a really great strategy.
You figured out that even numbers must go in the corners (you called them "edges," I think) and odd numbers must go in the spots beside them, like this:
We could tell that the odd had to be in the middle, because there are more odds than evens. Another clue was that 8 was given in the bottom left, and 8 is even.
You can actually prove that the odd numbers cannot go in the corners (that the following picture can't be true):
Recall that the sum of the numbers in a row, column or diagonal must add to 15 (this was in the video). However, the yellow column will have a sum that is even, because odd + odd + even = even. This is why the odd numbers can't go in the corners.
You found that 6 is in the bottom right, which is correct. The numbers you placed are slightly wrong, though, because 9 cannot be in the middle. Recall that the sum of the numbers in any row, column or diagonal should equal 15. If 9 were in the middle and 6 were in the bottom right, then since 9 + 6 = 15, then the third number (in the upper left box) would have to equal 0. However, we aren't allowed to use 0.
You had some very good ideas here, and that's what is important for developing real problem-solving skills!
The Daily Challenge Team
I forgot to answer your question about a different-shaped magic square. One reason the magic square must be a square is that since the sum of the numbers in each column, row or diagonal must equal 15, if you had a shape like a rectangle, there would be more numbers in a row than in a column, or vice versa. It wouldn't then be possible to get the same sum of numbers.
Great job thinking of this question!
The Daily Challenge Team
Sorry Sir - By 9 in the middle I meant 9 in the middle of the first row. However thank you for greatly explaining the concept.