Great exercises for your mind ---->

[bnashier]

1. You take a prism ( it is a regular solid with four faces where each face is triangular). You take four different colours, say - red, blue, green and white. You paint the four faces each with a different colour. How many essentially different colourings can you get? Convention: Two prisms have same colourings if you can place them (by rotating or any rigid motion) side by side so that they appear indentical.

2. A little more interesting. Now take a regular solid cube. It has six faces and each face a square. You take six different colours, say - red, blue, green, white, black and yellow. Paint the six faces each with a different colour. How many essentially different colourings can you get? Convention same as above.

Good luck and enjoy.

 

[amitrajora]

3 factorial and 5 factorial------?

 

[anujkumar]

1) Two
Explanation: Places the prism with bottom having red. Then rotate so that blue faces you. Now only to possibilities remain whether left back face is green or white giving two possibilities (answer could be one if you ignore the clockwise vs anticlockwise sense of the colors of three upper faces)
2)30
Place the red face as bottom. Then choose the color of top face (5 possibilities) Then rotate the square so that a given color faces you (one possibility) then remaining 3 faces can be colored in 6 different ways. So 6*5=30 possibilites.

Step one and three bascially fixes the orientation of the structure. any coloring which uses more that one possibility in these step are equivalant by rotation.

 

[bnashier]

Dear Anuj:

I am much impressed with your solutions. Correct indeed.

Anuj Kumar (Oct 02, 2003 02:36 a.m.):
1) Two
Explanation: Places the prism with bottom having red. Then rotate so that blue faces you. Now only to possibilities remain whether left back face is green or white giving two possibilities (answer could be one if you ignore the clockwise vs anticlockwise sense of the colors of three upper faces)
2)30
Place the red face as bottom. Then choose the color of top face (5 possibilities) Then rotate the square so that a given color faces you (one possibility) then remaining 3 faces can be colored in 6 different ways. So 6*5=30 possibilites.

Step one and three bascially fixes the orientation of the structure. any coloring which uses more that one possibility in these step are equivalant by rotation.