Another Pascal’s Triangle Gem

At a recent local conference, Jim Rubillo, Annie Fetter and I were saying our good-byes at the end of a fun evening, when Jim’s puzzly side emerged…

What proportion of the numbers in Pascal’s Triangle are even?

Every time I talk to Jim, he’s bound to have a neat problem for me to chew on.  The last time, he shared a fun task involving the harmonic series. Take a few minutes and think about this Pascal’s Triangle scenario…I’ll even leave you some spoiler space.

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At the ATMOPAV Spring Conference last month, Jim shared an entertaining talk titled “Gambling, Risk, Alcohol, Poisons and Manure – an Unfinished Life Story”.  The talk led the attendees on a journey through the history of statistics, starting with games of chance and the meeting of Chevelier du Mere and Blaise Pascal, through the introduction of formal inference procedures developed at the Guinness brewery, and to identifying statistical abuses in the present day.

Jim is a life-ling educator and former Executive Director of NCTM who happens to live quite close to me.  It was a thrill having him share his ideas with the group.

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Had enough time to think about this Pascal question?  Spoiler time is up!

So, which rows are in Pascal’s Triangle are we talking about here?

In theory, we are talking about “all” of the rows in the infinite Pascal’s Triangle, which makes this a bit tricky to think about for kids (and adults as well!).  But Jim shared with me slides which show the proportion of evens in increasing numbers of rows of the triangle.  You will notice that as the number of rows grows, the proportion of even entries also increases, and approaches 1.  What a neat result!  Below is an animated gif I made using a Pascal’s coloring applet which shows the increase in the proportion of even (white space) numbers in increasing rows.

For your class, this is a fun opportunity to talk about the parallels between Pascal’s Triangle, Sierpinski’s Gasket, and fractal area.

Already looking forward to my next encounter with Jim!