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Lego query (Posted on 2006-10-05) Difficulty: 4 of 5
Imagine you have a 4 x 2 blue lego, a 4 x 2 red lego and a 2 x 2 yellow lego. How many ways, excluding rotations, can you put the legos together (you must use all the pieces)?

No Solution Yet Submitted by joshua    
Rating: 3.2500 (4 votes)

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Some Thoughts A start - but far from the finish | Comment 1 of 7

Thoughts: With three blocks one block must be attached to the other two. Not ready to tackle that . . . yet.

Just going to go for two block connections. 74 varieties excluding rotations.

 

  My labeling:

 

Red     Blue   Yellow

 ___     ___    ____

|1 2|   |a b|   I  II

|3 4|   |c d|  III_IV

|5 6|   |e f|

|7_8|   |g_h|

 

Red to Blue:

One connection (4)

1 a

1 b

2 a

2 b

 

Two connections (8)

1 2 a b (respectively)

     a c

     c e

    e g

1 3 a b

    - a c

3 5 a b

5 7 a b

 

Three connections (2)

2 4 6 c e g

4 6 8 a c e

 

Four connections (5)

2 4 6 8 - a c e g

1 2 3 4 d c b a

        - b d a c

        - d f c e

        - f h e g

 

Five connections (0)

 

Six connections (1)

1 2 3 4 5 6 c d e f g h

 

Seven connections (0)

 

Eight connections (1)

1 2 3 4 5 6 7 8 a b c d e f g h

 

Now double the total since either the red or the blue block can be on top. 2(4 + 8 + 2 + 5 + 0 + 1 + 0 + 1) = 42

 

Yellow to Blue (or red)

One connection (2)

1 - I

2 I

 

Two connections (4)

1 2 I II

1 3 I II

3 5 I II

5 7 I II

 

Three connections (0)

 

Four connections (2)

1 2 3 4 I II III IV

3 4 5 6 I II III IV

Now double the total for top/bottom then double for red or blue connection to yellow 4(2 + 4 + 0 + 2) = 32

 

Next: Make a 42 by 32 matrix and eliminate the impossible combinations.

 

Edited on October 5, 2006, 5:03 pm
  Posted by Leming on 2006-10-05 16:55:18

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