[Guest]

I give you a table. Fill in the blanks so that the sum of each row should be identical, and sum of each column should be identical. But the sum of a column and row doesn't need to be same. You can use positive integers, except 0.

But my question isn't it. Of course it is easy to work on such a small table. But if I give you a table of 100x80 having a ladder shape blanks is here an easy way to fill it? I hope ladder is the true word to describe the form of the blanks in the table.

One of my friends used determinant way to fill it. But it is very hard for a greater table. Also giving variables to each blank is cumbersome.

Please don't get tired of filling my mini table, find a way to fill a greater table as 100x80, 100x150 etc.

9 1 7 4 _ _

3 6 2 _ _ 8

2 6 _ _ 7 2

8 _ _ 9 2 7

_ _ 8 1 6 3

[HoustonHokie]

Your table is 6x5 in dimension with two numbers missing in each row - is that right? I guess I'm having trouble seeing how you could have a similar "ladder" shape if the dimension were not X+1 x X. Conceptually, where would the blanks be in a 100x80 table, for example?

[Guest]

Your table is 6x5 in dimension with two numbers missing in each row - is that right? I guess I'm having trouble seeing how you could have a similar "ladder" shape if the dimension were not X+1 x X. Conceptually, where would the blanks be in a 100x80 table, for example?

Looks like a ladder then would need 21 blanks per row to fill the 100 x 80 grid. So we merely have 1,680 unknowns and 180 diophantine equations. Do we get infinite paper and pencil?

[Guest]

Looks like a ladder then would need 21 blanks per row to fill the 100 x 80 grid. So we merely have 1,680 unknowns and 180 diophantine equations. Do we get infinite paper and pencil?

Really, you and Hokie are quite right. Excuse me, I didn't think that. The ratio between cols and rows should be (x+1)/x, or reverse. I mean the ladder shape is essential and you has nearly row counts x 2 unknowns. I wonder only if you will see the easiest way to solve such ladder shape tables?

[Guest]

9 1 7 4 _ _

3 6 2 _ _ 8

2 6 _ _ 7 2

8 _ _ 9 2 7

_ _ 8 1 6 3

If you know the sums of rows and cols, it will be very easy, because once you fill the first col, the others will be filled by a ladder way.

The trick is that. Since there are 6 rows and 5 cols, and since total sum is constant then 5x(rows sum)=6x(cols sum). Now fill the first col so that sum of that col will be a multiply of 5, (nearest to logical is 25). Then fill the last row to complete to 30 (25x6/5). By the help of ladder shape, each blank will be filled. Although this is a simple problem, nobody before did solve it. (maybe I couldn't ask it throughly ) Only my old girl friend solved it by giving 0 to all sums. Since then, I banned the negative integers