put the number from 1 to 9 in the circles show that the sum of the numbers on each side of a triangle is the same
Answers
now try it..
Answer:
The numbers 1-9 add up to 45.
The sum of all three sides = 45 + (3 corners),
since each corner contributes to two sides.
Sum of all three sides = 3 * one side.
(3 corners) = sum of all three sides - 45.
Since both terms on the right side are divisible by 3,
the sum of the three corners is divisible by 3 too.
That limits the choices for the corners.
The sum of each side then will be (45 + (3 corners))/3.
From there, exhaustive search reveals the solutions below.
Each solution has a complement, where each number, k, is replaced by 10-k.
In the first two rows, the complements are one above the other,
In the bottom row, they are all self-complementary, with a reflection about the vertical axis.
The last pair are the only ones where, for a given choice for the corners, there is only one solution. The others are all in pairs.
Solutions
Side = 17
Side = 19
Side = 19
2
5 4
9 8
1 6 7 3
2
6 5
8 7
1 4 9 3
4
5 2
9 6
1 3 8 7
4
6 3
8 5
1 2 9 7
3
5 1
9 8
2 4 6 7
3
6 4
8 5
2 1 9 7
Side = 23
Side = 21
Side = 21
8
5 6
1 2
9 4 3 7
8
4 5
2 3
9 6 1 7
6
5 8
1 4
9 7 2 3
6
4 7
2 5
9 8 1 3
7
5 9
1 2
8 6 4 3
7
4 6
2 5
8 9 1 3
Side = 20
Side = 20
Side = 20
Side = 20
5
9 1
2 8
4 3 7 6
5
3 7
8 2
4 1 9 6
5
4 6
9 1
2 3 7 8
5
6 4
7 3
2 1 9 8
5
6 4
8 2
1 3 7 9
5
4 6
8 2
3 1 9 7