## Equal sums on lines

The first nine natural numbers (1, 2, 3, 4, 5, 6, 7, 8 and 9) are given. Draw two straight lines and place these numbers on these lines such that the sum of the numbers on each straight line is the same.

[SOLVED]

#### Pragya Shrivastava solved this puzzle:

One line containing 9 7 3 1 5 intersecting the line containing 8 6 4 2 5. 5 is the intersection point of the two lines. The sum of numbers in the both the lines is 25.

#### Susam Pal from cotpi solved this puzzle:

That's correct and that's just one of the many solutions possible. Here is a simple way to construct many such solutions. Draw two lines intersecting each other at a point. Place an odd number on the point of intersection. Let us call this number x. Now separate the remaining numbers into two sets such that the sum of the numbers in each set is (45 − x)2. Note that 45 is the sum the first 9 natural numbers. Place the numbers of one set on one line and the other set on the other line. The sum of the numbers in each line would be (45 + x)2 Here is another solution where 1 is at the point of intersection.

1 2 3 4 6 7 5 8 9

There are 31 such solutions possible which I have enumerated by writing a computer program. The format of the output is: [numbers in one line] [number at the point of intersection] [numbers in the other line].

1. [9 8 5] [1] [7 6 4 3 2]
2. [9 8 3 2] [1] [7 6 5 4]
3. [9 7 6] [1] [8 5 4 3 2]
4. [9 7 4 2] [1] [8 6 5 3]
5. [9 6 5 2] [1] [8 7 4 3]
6. [9 6 4 3] [1] [8 7 5 2]
7. [9 8 4] [3] [7 6 5 2 1]
8. [9 7 5] [3] [8 6 4 2 1]
9. [9 7 4 1] [3] [8 6 5 2]
10. [9 6 5 1] [3] [8 7 4 2]
11. [9 6 4 2] [3] [8 7 5 1]
12. [9 5 4 2 1] [3] [8 7 6]
13. [9 8 3] [5] [7 6 4 2 1]
14. [9 8 2 1] [5] [7 6 4 3]
15. [9 7 4] [5] [8 6 3 2 1]
16. [9 7 3 1] [5] [8 6 4 2]
17. [9 6 4 1] [5] [8 7 3 2]
18. [9 6 3 2] [5] [8 7 4 1]
19. [9 8 2] [7] [6 5 4 3 1]
20. [9 6 4] [7] [8 5 3 2 1]
21. [9 6 3 1] [7] [8 5 4 2]
22. [9 5 4 1] [7] [8 6 3 2]
23. [9 5 3 2] [7] [8 6 4 1]
24. [9 4 3 2 1] [7] [8 6 5]
25. [8 7 3] [9] [6 5 4 2 1]
26. [8 7 2 1] [9] [6 5 4 3]
27. [8 6 4] [9] [7 5 3 2 1]
28. [8 6 3 1] [9] [7 5 4 2]
29. [8 5 4 1] [9] [7 6 3 2]
30. [8 5 3 2] [9] [7 6 4 1]
31. [8 4 3 2 1] [9] [7 6 5]

### Credit

This puzzle is taken from folklore.