# Twelve Sum and Digit Substitution Puzzle

1. Aug 5, 2009

### K Sengupta

Substitute each of the capital letters in the figure given below by a different digit from 0 to 7 such that:

A+B+C = A+D+F = C+E+H = F+G+H = 12.

Code (Text):
A    B    C
D         E
F    G    H
Note: The rotations and reflections of a valid arrangement are deemed as the same solution.

2. Aug 5, 2009

### Edgardo

a=2 b=3 c=7 d=4 e=0 f=6 g=1 h=5 (solution 1)
a=2 b=4 c=6 d=3 e=1 f=7 g=0 h=5 (solution 2)
a=5 b=0 c=7 d=1 e=3 f=6 g=4 h=2 (solution 3)
a=5 b=1 c=6 d=0 e=4 f=7 g=3 h=2 (solution 4)
a=6 b=1 c=5 d=4 e=0 f=2 g=3 h=7 (solution 5)
a=6 b=4 c=2 d=1 e=3 f=5 g=0 h=7 (solution 6)
a=7 b=0 c=5 d=3 e=1 f=2 g=4 h=6 (solution 7)
a=7 b=3 c=2 d=0 e=4 f=5 g=1 h=6 (solution 8)

However, solutions 2-8 are reflections or rotations of solution 1.

(solution 1): a=2 b=3 c=7 d=4 e=0 f=6 g=1 h=5
2 3 7
4 _ 0
6 1 5

(solution 2): a=2 b=4 c=6 d=3 e=1 f=7 g=0 h=5
2 4 6
3 _ 1
7 0 5
(This is a reflection of (solution 1) along the AH-line)

(solution 3): a=5 b=0 c=7 d=1 e=3 f=6 g=4 h=2
5 0 7
1 _ 3
6 4 2
(This is a reflection of (solution 1) along the CF-line)

(solution 4): a=5 b=1 c=6 d=0 e=4 f=7 g=3 h=2
5 1 6
0 _ 4
7 3 2
(This is a rotation of (solution 1) by 180°)

(solution 5): a=6 b=1 c=5 d=4 e=0 f=2 g=3 h=7
6 1 5
4 _ 0
2 3 7
(This is a reflection of (solution 1) along the DE-line)

(solution 6): a=6 b=4 c=2 d=1 e=3 f=5 g=0 h=7
6 4 2
1 _ 3
5 0 7
(This is a rotation of (solution 1) by 90° to the right)

(solution 7): a=7 b=0 c=5 d=3 e=1 f=2 g=4 h=6
7 0 5
3 _ 1
2 4 6
(This is a rotation of (solution 1) by 90° to the left)

(solution 8): a=7 b=3 c=2 d=0 e=4 f=5 g=1 h=6
7 3 2
0 _ 4
5 1 6
(This is a reflection of (solution 1) along the BG-line)