# Squares, subsquares and arrangement puzzle

1. Aug 6, 2009

### K Sengupta

Substitute each of the capital letters in this 3x3 square by a different digit from 1 to 9 such that the sum of digits in each of the four 2x2 subsquares is equal to 6*E.

A B C
D E F
G H I

What will be the arrangement(s), if keeping all the other conditions unaltered, the sum of digits in each of the four 2x2 subsquares is equal to 5*E?

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

2. Aug 8, 2009

### Edgardo

Part 1 of the puzzle:
There are 8 solutions, however solution 2-8 are either reflections
or rotations of solution 1.

solution1: a=4 b=2 c=7 d=9 e=3 f=6 g=5 h=1 i=8
solution2: a=4 b=9 c=5 d=2 e=3 f=1 g=7 h=6 i=8
solution3: a=5 b=1 c=8 d=9 e=3 f=6 g=4 h=2 i=7
solution4: a=5 b=9 c=4 d=1 e=3 f=2 g=8 h=6 i=7
solution5: a=7 b=2 c=4 d=6 e=3 f=9 g=8 h=1 i=5
solution6: a=7 b=6 c=8 d=2 e=3 f=1 g=4 h=9 i=5
solution7: a=8 b=1 c=5 d=6 e=3 f=9 g=7 h=2 i=4
solution8: a=8 b=6 c=7 d=1 e=3 f=2 g=5 h=9 i=4

solution1: a=4 b=2 c=7 d=9 e=3 f=6 g=5 h=1 i=8
4 2 7
9 3 6
5 1 8

solution2: a=4 b=9 c=5 d=2 e=3 f=1 g=7 h=6 i=8
4 9 5
2 3 1
7 6 8
(reflection of solution 1 along the AI-line)

solution3: a=5 b=1 c=8 d=9 e=3 f=6 g=4 h=2 i=7
5 1 8
9 3 6
4 2 7
(reflection of solution 1 along the DF-line)

solution4: a=5 b=9 c=4 d=1 e=3 f=2 g=8 h=6 i=7
5 9 4
1 3 2
8 6 7
(rotation of solution 1 by 90° to the right)

solution5: a=7 b=2 c=4 d=6 e=3 f=9 g=8 h=1 i=5
7 2 4
6 3 9
8 1 5
(reflection of solution 1 along the BH-line)

solution6: a=7 b=6 c=8 d=2 e=3 f=1 g=4 h=9 i=5
7 6 8
2 3 1
4 9 5
(rotation of solution 1 by 90° to the left)

solution7: a=8 b=1 c=5 d=6 e=3 f=9 g=7 h=2 i=4
8 1 5
6 3 9
7 2 4
(rotation of solution 1 by 180°)

solution8: a=8 b=6 c=7 d=1 e=3 f=2 g=5 h=9 i=4
8 6 7
1 3 2
5 9 4
(reflection of solution 1 along the CG-line)

3. Aug 8, 2009

### Edgardo

Part 2 of the puzzle:
Part 2 of puzzle:
With the 5*E condition we get 16 solutions:
solution1: a=1 b=6 c=7 d=9 e=4 f=3 g=2 h=5 i=8
solution2: a=1 b=9 c=2 d=6 e=4 f=5 g=7 h=3 i=8
solution3: a=2 b=5 c=3 d=9 e=4 f=8 g=6 h=1 i=7
solution4: a=2 b=5 c=8 d=9 e=4 f=3 g=1 h=6 i=7
solution5: a=2 b=9 c=1 d=5 e=4 f=6 g=8 h=3 i=7
solution6: a=2 b=9 c=6 d=5 e=4 f=1 g=3 h=8 i=7
solution7: a=3 b=5 c=2 d=8 e=4 f=9 g=7 h=1 i=6
solution8: a=3 b=8 c=7 d=5 e=4 f=1 g=2 h=9 i=6
solution9: a=6 b=1 c=7 d=9 e=4 f=8 g=2 h=5 i=3
solution10: a=6 b=9 c=2 d=1 e=4 f=5 g=7 h=8 i=3
solution11: a=7 b=1 c=6 d=8 e=4 f=9 g=3 h=5 i=2
solution12: a=7 b=3 c=8 d=6 e=4 f=5 g=1 h=9 i=2
solution13: a=7 b=6 c=1 d=3 e=4 f=9 g=8 h=5 i=2
solution14: a=7 b=8 c=3 d=1 e=4 f=5 g=6 h=9 i=2
solution15: a=8 b=3 c=7 d=5 e=4 f=6 g=2 h=9 i=1
solution16: a=8 b=5 c=2 d=3 e=4 f=9 g=7 h=6 i=1

However, the only "real" solutions are solution 1 and 3.
Solution 2,4,5,12,13,15,16 are either reflections or rotations of solution 1.
Solution 6,7,8,9,10,11,14 are either reflections or rotations or solution 3.

solution1: a=1 b=6 c=7 d=9 e=4 f=3 g=2 h=5 i=8
1 6 7
9 4 3
2 5 8

solution2: a=1 b=9 c=2 d=6 e=4 f=5 g=7 h=3 i=8
1 9 2
6 4 5
7 3 8
(reflection of solution 1 along the AI-line)

solution3: a=2 b=5 c=3 d=9 e=4 f=8 g=6 h=1 i=7
2 5 3
9 4 8
6 1 7
(neither rotation nor reflection of solution 1)

solution4: a=2 b=5 c=8 d=9 e=4 f=3 g=1 h=6 i=7
2 5 8
9 4 3
1 6 7
(reflection of solution 1 along the DF-line)

solution5: a=2 b=9 c=1 d=5 e=4 f=6 g=8 h=3 i=7
2 9 1
5 4 6
8 3 7
(rotation of solution 1 by 90° to the right)

solution6: a=2 b=9 c=6 d=5 e=4 f=1 g=3 h=8 i=7
2 9 6
5 4 1
3 8 7
(neither rotation nor reflection of solution 1)
(but reflection of solution 3 along the AI-line)

solution7: a=3 b=5 c=2 d=8 e=4 f=9 g=7 h=1 i=6
3 5 2
8 4 9
7 1 6
(neither rotation nor reflection of solution 1)
(but reflection of solution 3 along the BH-line)

solution8: a=3 b=8 c=7 d=5 e=4 f=1 g=2 h=9 i=6
3 8 7
5 4 1
2 9 6
(neither rotation nor reflection of solution 1)
(but rotation of solution 3 by 90° to the left)

solution9: a=6 b=1 c=7 d=9 e=4 f=8 g=2 h=5 i=3
6 1 7
9 4 8
2 5 3
(neither rotation nor reflection of solution 1)
(but reflection of solution 3 along the DF-line)

solution10: a=6 b=9 c=2 d=1 e=4 f=5 g=7 h=8 i=3
6 9 2
1 4 5
7 8 3
(neither rotation nor reflection of solution 1)
(but rotation of solution 3 by 90° to the right)

solution11: a=7 b=1 c=6 d=8 e=4 f=9 g=3 h=5 i=2
7 1 6
8 4 9
3 5 2
(neither rotation nor reflection of solution 1)
(but rotation of solution 3 by 180°)

solution12: a=7 b=3 c=8 d=6 e=4 f=5 g=1 h=9 i=2
7 3 8
6 4 5
1 9 2
(rotation of solution 1 by 90° to the right)

solution13: a=7 b=6 c=1 d=3 e=4 f=9 g=8 h=5 i=2
7 6 1
3 4 9
8 5 2
(reflection of solution 1 along the BH-line)

solution14: a=7 b=8 c=3 d=1 e=4 f=5 g=6 h=9 i=2
7 8 3
1 4 5
6 9 2
(neither rotation nor reflection of solution 1)
(but reflection of solution 3 along the CG-line)

solution15: a=8 b=3 c=7 d=5 e=4 f=6 g=2 h=9 i=1
8 3 7
5 4 6
2 9 1
(reflection of solution 1 along the CG-line)

solution16: a=8 b=5 c=2 d=3 e=4 f=9 g=7 h=6 i=1
8 5 2
3 4 9
7 6 1
(rotation of solution 1 by 180°)