Squares, subsquares and arrangement puzzle

In summary, squares are two-dimensional shapes with four equal sides and four right angles, while subsquares are smaller squares contained within a larger square. These shapes can be used in arrangement puzzles where the player must rearrange them in specific ways for the purpose of improving spatial reasoning, critical thinking, and problem-solving skills. Famous examples of such puzzles include the Tangram and Soma cube.
  • #1
K Sengupta
113
0
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.
 
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  • #2
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
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°)
 

1. What is a square?

A square is a two-dimensional shape with four equal sides and four right angles.

2. What are subsquares?

Subsquares are smaller squares that are contained within a larger square. They can be created by dividing the larger square into equal parts.

3. How can arrangement puzzles involve squares and subsquares?

Arrangement puzzles can involve squares and subsquares by challenging the player to rearrange a set of squares and subsquares in a specific way, such as creating a larger square using smaller subsquares or arranging a set of squares to create a specific pattern.

4. What is the purpose of solving arrangement puzzles involving squares and subsquares?

Solving arrangement puzzles involving squares and subsquares can help improve spatial reasoning skills, critical thinking abilities, and problem-solving skills.

5. Are there any famous examples of arrangement puzzles involving squares and subsquares?

Yes, the well-known Tangram puzzle involves arranging seven flat shapes, including a square and five triangles, to form a larger square. The Soma cube puzzle is another example, which involves arranging seven pieces to create a 3x3x3 cube.

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