Based on a reading of Google searches of visitors to this site, here are a few of the more common topics, rephrased as questions, that I have noticed. I also noted that, aside from pi day, which generated hundreds of visits from those looking for historical information about pi and the names of mathematicians, many are looking for sample MathCounts problems or references to books of these.

1. What is the largest 3-digit prime with 3 prime digits?

Ans: 773 if repeated digits are allowed; 523 if not. This question has appeared in one of my earlier posts. Must be a fairly common one that readers come across from math contests or from class. A quick list of primes up to 1000 can be found at VIAS Encyclopedia.

2. How are the following questions related:

How many different handshakes occur if each of 10 people shakes hands with each of the remaining people in a room? (can be expressed more clearly)

How many different segments can be formed by connecting the vertices of a decagon in all possible ways? (can also be expressed in terms of the number of chords formed by 10 points on a circle)

Ans: Both problems can be expressed as 9+8+7+6+...+2+1 or (9)(10)/2 or 10C2, the number of combinations of 10 objects chosen 2 at a time. In general, for n people or points on a circle: 1+2+3+4+...+(n-1) = (n-1)(n)/2 = nC2. The equivalence comes from the fact that each handshake or each chord is uniquely determined by selecting 2 people or 2 points. To avoid repetition we use combinations rather than permutations.

4. How many different seven-digit phone numbers (ignoring the area code) can be formed?

Ans: If any digit 0-9 were allowed in all positions then there would be 10^{7} possibilities since there are 10 choices for each digit (using the Multiplication Principle here). Subject to restrictions on zeros or other digits or other local considerations there would be fewer.

Here's a related question that I have not yet seen:

How many different IP addresses are possible worldwide if every computer, device etc., must have a unique one?

I believe that IP addresses are always of the form xxx.xxx.xxx where the first digit in each group is allowed to be zero, that is, one could use a 2-digit number in each group. For example, 66.19.35 is acceptable. I didn't research the rules so this may be incorrect.

Ans: If the rule of formation is accurate, there could be 10^{9} or one billion IP addresses. How long will it take for these to be used up? I know someone out there will tell us how this is or will be handled!

Update on IP Addresses: My belief about the form of IP addresses was dead wrong! The protocol should have been 4 groups of integers, each in the range from 0 through 255. There are now newer protocols to allow for the exponential growth of devices needing an address. See the comments for this post to learn more from those far more knowledgeable than myself!

## Tuesday, March 20, 2007

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Posted by Dave Marain at 5:35 AM

Labels: combinatorial math

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## 3 comments:

The problems with IP addresses have already been answered. The scheme you are referring to, technically known as IPv4, is being replaced with the new IPv6. IPv6 addresses are of the form HHHH.HHHH.HHHH.HHHH… [8 groups in all], where each H is a hexidecimal digit (though there are ways to shorten the string when some of the digits are zero). Thus, there are 2^{128} addresses. This should be enough for a while.

Wikipedia also explains that the designation IPv5 was already in use for another protocol.

For the IP address question... actually, there are four octets (xxx.xxx.xxx.xxx) but each octet ranges in value (base 10) from 0 to 255. So the smallest IP address would theoretically be 0.0.0.0 and the largest/highest would be 255.255.255.255. So there are theoretically 256^4, or roughly 4.29 billion.

You probably don't want to worry about the fact that there are certain non-routable ranges (like 192.168.xxx.xxx) that can't be used out on the "public" Internet.

What's most amazing to me is that we're actually running out of IPv4 addresses!!!! But with all of our IP-addressable devices (not just traditional computers) one person can "use up" a good number of IP addresses!

eric, rich--

thank you so much for educating me about the different protocols! i should have recognized the 0-255 restriction; the new iPv6 scheme which produces 16^32 = 2^128 possible addresses should be enough for a few millennia! let's see, if there are about 4 trillion or 4x10^12 people in the world and they each need...

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