The four numbers A, B, A+B and A-B are all prime. The sum of these four numbers is

A) Even

B) Divisible by 3

C) Divisible by 5

D) Divisible by 7

E) Prime

Source: 2002 AMC 10/12B #15

The four numbers A, B, A+B and A-B are all prime. The sum of these four numbers is

A) Even

B) Divisible by 3

C) Divisible by 5

D) Divisible by 7

E) Prime

Source: 2002 AMC 10/12B #15

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I was having a problem when using Camtasia Studio to do a screen capture of an algebra lecture. On my laptop, the capture works just fine but on my desktop it was very jumpy. For those who don’t know, I use a Wacom Tablet, Microsoft OneNote and Camtasia to produce a series of videos for our online Algebra sequence.

Whenever I would begin recording, the system would slow down enough that the writing on the screen was broken and hard to read. The sound capture was fine, it just seemed that the CPU was not able to keep up with capturing the video on the screen and allow me to write smoothly. The laptop, where it works fine, is a faster processor but with the same amount of memory. I’m not certain how the video adapters compare.

I first discovered the problem several months ago and had been switching back and forth ever since. However, today I took initiative and attempt to solve the problem one more time and came across a tip I had not considered.

The Solution that worked for me: Reduce the color depth from 32 bit to 16 bit. For the types of videos I am doing, that makes makes no discernible difference and now it is as smooth on my desktop as it is on my laptop.

Some other tips for increasing the capture rate were found here: **http://tinyurl.com/arua4b**

Here’s a short sample:

I love these kinds of puzzles because there is only one solution but there are several ways to get there. It’s at least as interesting to hear the different approaches as it is to solve it.

Here was the puzzle posted last week:

Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with 4-digit number in which each of two digits appears two times.

“Look, daddy!” she exclaims. “That number is evenly divisible by the age of each of us kids!”

“That’s right,” replies Mr. Jones, “and the last two digits just happen to be my age.”

What is the four digit number in the license plate?

The first thing I noted when I saw the problem is that the solution must be in one of the following forms: aabb, abba, or baba (where a and b are distinct digits). Also, I noticed that since the oldest child is 9, the sum of digits must be a divisible by 9. So that 2a + 2b = 9k for some k. Because 2a+2b is even, k must be even. We know that a and b are distinct digits between 0 and 9, inclusive, the largest 2a+2b could be is 34, so k must be 2. In other words, a+b=9.

Next thing I noticed is that Mr. Jones will have either 4 year old or an 8 year old or both. Either way, the 4-digit number is divisible by 4. This means the last two digits must be divisible by 4. We thus have the following possibilities: 9900, 5544, 1188, 3636, 6336, 2772, 7272. I arrived at these by thinking first of numbers of the form bb that are divisible by 4 (00, 44, 88) and providing the appropriate a to get aabb that is divisible by 9. Then I considered the numbers of the form ba, that are divisible by 9 and 4 (36, 72) and listed both abba and baba.

Notice that in this list only 9900 is divisible by 5 and 00 can’t possibly be the age of Mr. Jones so he must not have a child age 5. Only 5544 is divisible by the digits 1, 2, 3, 4, 6, 7, 8, and 9.

SOLUTION: 5544

A few folks emailed me their solutions and were all correct, but each had an approach that was not quite the same as mine. Well done to those!

An awesome new feature was recently announced by Google for the contact management tools in Gmail. If you have multiple contact entries for the same individual in Gmail you can now easily merge them into one.

(From Lifehacker)

For example, if you’re staring in the face of numerous duplicate contacts that should represent the same person, the built-in contact merge feature in Google Contacts is a must. Just find the duplicate contacts, tick their checkboxes, and click "Merge these contacts…." Easy peasy. To manage your contacts, either head to the Contacts page in Gmail or to the unadvertised standalone site.

Gmail just keeps getting better. I was back to using Outlook for long while until the Tasks feature was launched. Now, all my email addresses are dumped into Gmail. I don’t think I’m going back.

Oh, and thanks to the IMAP capabilities in Gmail, I have uploaded all my archived email into my Gmail account. I can search my work emails dating all the way 2002.

Oh, and I love the new Multiple Inboxes feature, as well.

Oh, and how about those themes?

You’ll find nothing but love for Google here…

Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with 4-digit number in which each of two digits appears two times.

“Look, daddy!” she exclaims. “That number is evenly divisible by the age of each of us kids!”

“That’s right,” replies Mr. Jones, “and the last two digits just happen to be my age.”

What is the four digit number in the license plate?

Source: 2006 AMC 10/12B #25 (American Mathematics Contest)

Two days ago, I posted this simple little number puzzle. Quite a few folks came up with the answer below. One of the interesting questions you can ask is whether that solution is unique.

Clearly there are two lines of symmetry in the original problem so by reflection alone we come up with a total of four solutions: [tex] { I, F_x, F_y, F_x circ F_y } [/tex] where [tex]F_x [/tex] and [tex] F_y [/tex] represent “flips” across the lines of symmetry and [tex] I [/tex] represents the identity, or the solution above. By [tex]F_x circ F_y[/tex], I mean the composition of the flipping operations or just consecutive flipping.

There also exists radial symmetry at [tex]180^{circ}[/tex], but this is equivalent to [tex]F_x circ F_y[/tex]. So for this arrangement above, there are four solutions of the same “type”.

Are any other arrangements possible besides these four?

I’m not keeping very good track of how many of these kind of puzzles I’m posting, so we’ll just say this is the 6th.

**Problem**: Place the digits 1 through 8 in the circles below such that no two adjacent circles contain consecutive digits.

*Update: I was missing a couple of lines. The picture is now correct.*