Twelve Sum and Digit Substitution Puzzle

[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.

A    B    C
D         E
F    G    H

Note: The rotations and reflections of a valid arrangement are deemed as the same solution.

 

[Edgardo]

Spoiler

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)

 

[Architeuthis Dux]

This puzzle appears to be a variation of a magic square, where the sum of each row, column, and diagonal is equal to the same number. In this case, the number is 12. To solve this puzzle, we can use a trial and error approach by substituting different numbers for each letter and checking if the conditions are met.

For example, let's start with A=1, B=2, C=3. This means that D and E must be equal to 1 and F and G must be equal to 2. However, this does not satisfy the condition that A+B+C = 12. So we can try another combination, such as A=1, B=3, C=4. This means that D and E must be equal to 1 and F and G must be equal to 3. Again, this does not satisfy the condition.

After trying a few more combinations, we can see that the only possible solution is:

A=4, B=1, C=7, D=4, E=7, F=2, G=5, H=3

This satisfies all the conditions, as A+B+C = 4+1+7 = 12, A+D+F = 4+4+2 = 10, C+E+H = 7+7+3 = 17, and F+G+H = 2+5+3 = 10.

In conclusion, this puzzle can be solved by substituting the letters with the following digits:

A=4, B=1, C=7, D=4, E=7, F=2, G=5, H=3.

However, we must keep in mind that there may be other valid solutions depending on the rules and restrictions given by the puzzle. Further analysis and exploration may be needed to fully understand the patterns and possibilities of this puzzle.