The Math Factor Podcast

We’ve never discussed the famous “Monty Hall Problem” here (though we did talk about it on the radio before we started podcasting). We recently got an interesting letter that highlights the difference between a game like “Let’s Make A Deal” and a game like “Deal or No Deal”.

Mark A. recently wrote us:
Read the rest of this entry »

I had a dream last night involving — (?) well I am not really sure, except that it left me wondering if there is a simple proof (if indeed it is true) that there must be a common factor of

m choose i = m!/(i! (m-i)!)
m choose j = m!/(j! (m-j)!)

for all counting numbers i,j,m with 1 < i,j < m

Another way to state this same thing is: any pair of entries, on any row of Pascal’s triangle (except for the 1’s on the edges) will have a common factor.

With facts of this sort, often there is a clever way to cast things in terms of counting something a couple of different ways which makes things clear.

Steve D. wrote us to say:

I was listening to another podcast and they misread the copy and ended up
saying “What is the most numerous number?”. Well, what IS the most numerous number?

This is really a fascinating question! Have you ever wondered, for example, why there are 7 of so many things:

• 7 wonders of the ancient world
• 7 mortal sins
• 7 stars in the big dipper
• 7 days of the week
• 7 dwarves
• 7 brides for 7 brothers
• 7 items on this list

Really, it’s not that big of a mystery. The fact is, small numbers are very useful, and get called upon a lot. But there aren’t that many of them to go around.

Hence, the First Strong Law of Small Numbers: There aren’t enough small numbers to meet the many demands placed upon them!

The most numerous numbers, in a sense then, are the small ones. Google searches seem to confirm this:

• 1, 2, 3, 4, … (several billion hits each)
• 78, 122, 157, … (several hundreds of millions of hits each)
• 12122…(millions of hits)
• 1278232… (hundreds of hits)

Lotsa fun can be had in this way… With a little fishing, you can find some ridiculously large numbers with more hits than they deserve, but the principle is clear.

This same principle, incidentally, explains why, for example, the Golden Ratio appears in so many settings. There’s nothing really that mystical about it. The Golden Ratio is a root of the polynomial x²-x-1=0. Roots of polynomials come up all over the place, in countless applications. And just as small numbers are in great demand, roots of simple polynomials will appear over and over again.

The Golden Ratio is just about the simplest non-integer root possible, and so, of course, shows up endlessly.

Challenge Question I’m kind of curious now: What is the smallest counting number that is NOT on the web?

210210876 was not on the web until just now, according to Google. Internet history has just been made!! But I’m sure you can find something smaller…

We never did resolve the question of which grows faster:

In this corner we have
Sequence 1 n^^n
1, 2^2, 3^3^3, 4^4^4^4, and so on.

And over here we have Sequence 2, defined recursively by

In short, we can define the second sequence as s(1) = 1; s(n) = n, followed by s(n-1) factorial signs.

Which sequence grows faster than the other??

We have many conflicting answers, and no decisive resolution; here was one idea .

[[[[This post has been corrupted somehow and we will try to get around to repairing it soon]]]]

David R. of Palo Alto writes:

Have you ever discussed factorials on your podcast? I don’t recall,
but a friend and I are puzzled and so of course we turn to you: Why
is “zero factorial” 1? Was it simply defined that way to frustrate
all of us nonmath folks, or is there a valid explanation?

Read the rest of this entry »