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Hi:
I need help finding how many numbers are divisible by 3 in the nth row of Pascal's Triangle. I have studied pascal's triangle and looked for patterns but am unable to find a pattern. Could someone please shed some light on my work?

Thanks

Vikas

Vikas,

Can you please provide a little more information about the work you are doing? How is this question related to your science fair project? If you do an internet search for Pascal's Triangle you'll be directed to a number of resources that should help you understand the patterns that can be found. Once you understand the patterns and the equations related to Pascal's Triangle you'll likely have a clearer answer to your question.

Please post a follow-up question after your research if you are still having trouble.

Regards,
Melissa G.
Science Buddies Staff

Thank you for responding to my post.

I came across this problem when I was researching Pascal's triangle, I have not yet decided on a project. I have narrowed it down to Pascal's triangle and maybe something relating to Serpinski Triangle. I was doing this problem just as research, for understanding. I have surfed the web a lot already but nothing has come up on dividing any row by 3. I need some help on seeing the pattern.

Vikas

If you just want to be able to visualize the pattern, there's a wonderful little tool at http://www.cut-the-knot.org/Curriculum/ ... ngle.shtml that shows you a pictorial representation of Pascal's Triangle modulo arbitrary numbers (e.g. a modulus of 3 diagram would have white circles where the number is divisible by 3).

There are certainly many obvious patterns (for example, the 3^Nth rows contain numbers that are all multiples of 3, other than the 2 1s on the ends, of course, and the 3^Nth-1 rows all have *no* numbers that are multiples of 3), but I'm not sure there's a simple equation that would define it for all rows. Based on some information in these sources, I suspect that it will tend to be on the order of the logarithm of the row number, but with wild fluctuations.

In the flavor of this forum, certainly the easiest way to find one of these values would be to write a little program that calculates it. There's a formula for calculating all the values of the Nth row of Pascal's Triangle in the wikipedia entry for it (for example), so you can even do it for an arbitrary row if you like, rather than generating all of them. It's possible that if you stare at that formula long enough you might come up with a way to represent this in a closed form.

is anyone familiar with PASCAL PYRAMID??

rappy,

I have posted a reply to your thread on Pascal's Triangle. For future references, please put your questions on separate threads in order to keep the forum organized. Thank you!