Coins & Lines

Probability Level 4

Number of "different" ways to place 10 identical coins in such a way that they lie in 5 straight lines & on each line there are exactly 4 coins.

NOTE : Here "different" means do not consider an arrangement that is rotated slightly as a completely different way.


The answer is 2.

Gaurish Korpal by Gaurish Korpal , 25, India

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2 solutions

Rama Devi
May 25, 2015

This is one of the problems that are there in the Shakuntala devi puzzles to puzzle you book.

Therefore the required answer is 2

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Gaurish Korpal
Mar 26, 2014

For solution see figures - Alternate text Alternate text

2 Helpful 0 Interesting 0 Brilliant 0 Confused

How do you know that those are the only 2 methods?

Also, what makes the 2 ways different? In the star, if I were to shift the horizontal line down, I will get the A-frame answer.

Calvin Lin Staff - 7 years, 2 months ago

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I was inspired form this problem from my high school course - "Computer Networking" where we have to find all possible networks by "Hit & trial" . From my experience in this "field" I know that there are two "different" ways only,( if you don't move connectors, as you have suggested).

You may disprove me by suggesting a third way, it would be a learning part for me!

Gaurish Korpal - 7 years, 2 months ago

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It would depend on what you mean by "different" ways. It has to be defined up to some idea of "same" (i.e. isomorphism), which was not clarified in your question. For example, you do not want to consider a star that is rotated slightly as a completely different way.

I labelled the vertices - Image Image

There is a graph isomorphism where the black vertex is mapped onto the blue vertex with the corresponding number. In this way, the lines are 1 , 3 , 6 , 9 1,3,6,9 and 1 , 4 , 7 , 10 1,4,7,10 and 2 , 3 , 4 , 5 2,3,4,5 and 2 , 6 , 8 , 10 2,6,8,10 and 5 , 7 , 8 , 9 5,7,8,9 .

Calvin Lin Staff - 7 years, 2 months ago

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@Calvin Lin – I edited my post. Are you satisfied now?

Gaurish Korpal - 7 years, 2 months ago

i cant understand the problem, by me points of intersection of 5 non parallel , non concurrent lines would provide the required ten points satifying the given condition ,,,, so there must be an infinite set of such lines.

Abhay Agarwal - 7 years ago

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