## Interesting little Math problem

Here's an interesting little problem. I made this one up but I bet it's been thought of by students of number theory and diophantine equations in particular.

Let's say you have two kinds of postage stamps, one with a value of 3 cents, the other with a value of 5 cents. List the postage amounts that you CAN'T make. You can't make 1, 2, 4, or 7 cents.

Let's look at another example. You have a 5 cent stamp and an 8 cent stamp. What values can't you make? You can't make 1, 2, 3, 4, 6, 7, 9, 11, 12, 14, 17, 19, 22, 23, or 27 cents.

For 3 and 5 cent stamps, 7 cents is the largest amount you can't make.

For 5 and 8 cent stamps, 27 cents is the largest amount you can't make.

Can you come up with a simple formula for the largest amount you can't make given two kinds of stamps? What assumption must you make in order for there to be a largest amount you can't make?

Can you explain why your simple formula works?

## What patterns can you find in Pascal’s triangle?

Pascal's triangle is a very simple thing to construct, yet it has a tremendous amount of depth to it. Also, Pascal's triangle has a huge number of interesting patterns in it. If you're not familiar with Pascal's triangle then read this background article at Wikipedia and then try your hand at finding some of the following patterns in the numbers, or in sums of numbers, or in some other relationship among numbers. The patterns might appear in a line going across the triangle, along one of the diagonals or elsewhere.

There are a couple of resources listed at the end of this article that give a number of the answers but try to find them yourself before checking the references.

## How to stumble upon great Math websites

I've been using StumbleUpon for a few months now. StumbleUpon is a free social networking community like Digg, Delicious and many others. What's unique about StumbleUpon is that you download a toolbar that lets you stumble upon (view) random sites on many different areas of interest. When you download the toolbar you get a stumble button. If you've configured your StumbleUpon account to show you Math-related sites then every time you click that stumble button you'll get a new Math site.

What's particularly nice about surfing the web with StumbleUpon is that the sites are not quite randomly selected, even in your field(s) of interest. When you "StumbleUpon" a site you can click the "thumbs up" or "thumbs down" button (part of the toolbar) to vote for the site. You can also vote for sites you discover on your own. The StumbleUpon system takes user votes into account when showing you new web pages. So, if someone submits a site to the system (by going to that web page and clicking the StumbleUpon button) then the site is shown to some people. If enough people give the site a thumbs up then it is shown to more people. So, sites StumbleUpon shows you are often ones that have received a large number of positive votes.

## Math magic with the number 9

9 is a most interesting number. I'm sure that's largely because 9 is 1 less than 10 and most of us have 10 fingers (or digits) and we do arithmetic in a base 10 system. I've seen an amazing number of math tricks that take advantage of something called "digital roots", which is closely related to the idea of "casting out nines." I want to introduce you to these two concepts and share some fun Math tricks you can do with this "9 stuff."

The digital root of a number, and this only makes sense for whole numbers, is what you get when you add up all of its digits. So, the digital root of 112 is 1+1+2, or 4. The digital root for 1234 is 1+2+3+4, which is 10. Now, when you're computing digital roots you only want a single digit so in the case of 1234, you add up its digits to get 10 then add 1+0 to get 1. So, 1 is the digital root of 1234.