The first solution was sent in by Jacob. He made four rectangles.

The second solution was submitted by Andrei , School 205 Bucharest, who was able to make five rectangles but one of them contains only two numbers.

Can anyone improve on five or offer a different solution with 5 rectangles?

Is it possible to to distribute the numbers so there are no rectangles totalling 34, or just one, two or three?

Jacob's solution:

5 13 17
15 1 3
9 11
7 19
5 13 17
15 1 3
9 11
7 19
5 13 17
15 1 3
9 11
7 19
5 13 17
15 1 3
9 11
7 19


Andrei's solution:
First, I put number 19 in the top left-hand corner, because it is the biggest. I see that adding 15 to 19 gives 34, the number I should obtain. So, I place 15 in the opposite corner, in the top right.

19 15

Now, I see that

19 + 11 + 1 + 3 = 34

I place the numbers as follows:

19 15
1
3
11

I see that

34 - (19 + 1) = 14

and

14 = 9 + 5

I place the numbers in the big rectangle:

19 15
1 9 5
3
11

Now, I apply the same method, seeing that

11 + 3 = 9 + 5

and

34 - (13 + 7) = 11 + 3

19 15
1 9 5
3
11 13 7
19 15
1 9 5
3
11 13 7



I did not use 17, so I put it in the corner that was left:

19 15
1 9 5
3
11 13 7 17

In total I found five rectangles.