1. Make a grid of 0’s.  Really you just need a border of 0’s, but filling an area is faster.  It should be twice as wide as tall.
2. Select cell B2, and give it the formula =MOD(A1+c1,10).
   
3. Copy B2 and paste it into all the cells in the grid except for the top row and the left and right columns.
4. Now they are all adding the values of the two cells diagonally above them.  Just put a “1″ in the middle of the top row and see what happens.
   
5. To color the different digits, click Format, Conditional Formatting, and make some stuff up.
   
6. I changed the font color to white and clicked File > Web Page Preview:
   

You could do some fun experiments with this, leading kids toward the idea of cellular automata!

A pack of zombie dogs (ZDs) is reproducing quickly and killing humans. Scientists and calculus students have determined that the the total number of human deaths due to ZDs t days after the start of 2010 is given by

1. At the end of January 1st (t = 1), how many deaths have occurred?

2. At the end of January, how many deaths will have occurred?

3. Determine the number of deaths/day after t days: h′(t).
Hint: you may want to go “outside-in” with the chain rule.

4. Graph both h(t) and h’(t) on the same axes:

5. On what day will 90% of the human population (6.8 billion) be annihilated by ZDs?

6. When does the death rate reach half a billion per day?

7. When does the death rate hit its peak?

Can you match them all? Height of a projectile, iTunes sales, World Population of gorillas, Transfer speed versus message size on several networks, Heights of a group of men, Population of fruit flies in a container, Decay of radioactive atoms, Heights of babies over time.

If you want to play it, here it is.

1. Teacher Soundboard
   Add  random sound effects to your instruction.  Negligible educational value.
   
2. Brown Blobs
   Knock the brown spots off the graph with an appropriate linear equation.
   Let me know what happens when you get to level 100 – no one has gotten there yet.
   
3. Golden Spirals
   Explore the sunflower’s perfect seed arrangement.
   
4. Trig Explorer
   Examine the relationships between the three trig functions.
   
5. Slope Explorer
   Guess the slope of a random line segment.
   
6. G-Map Concept Mapper
   Quickly generate a concept web
   
7. Classroom Timer
   Just a huge timer to show on your screen.
   

Found at http://www.geohive.com/earth/gen_popsize.aspx

Jorge Luis Borges‘ short story Library of Babel conceives of a library comprised of all possible books.  Specifically, all possible permutations of 25 characters (22 letters, comma, period, space) within books with 410 pages.  (Each page has 40 lines; each line has 80 characters).  This set of books would contain all books written so far (spanning multiple volumes if needed), along with all books that could possibly be written in the future.  The majority of the books in the library would be gibberish, of course.

A few questions, ordered from easy to hard:

• How many books are in The Library of Babel?
• How much space would this set of books consume (let’s ignore the rooms and shelves, and just stack the books, which measure, say, 6 in. x 4 in. x 1.5 in. )?
• What proportion of the books would read coherently from start to finish?

In this library, there would be a book explaining the solutions to each of these questions!

1. Neal Ziring’s Mandelbrot Applet

Pros: Easy navigation, anti-aliasing, control over the color gradient.
Cons: I couldn’t get it to save the image (had to PrtScn), and it aborted computation when I selected a large image size.

2. Mandelbrot Fractal Explorer

Pros: Good navigation controls (double-click or shift+drag to zoom, drag to pan), lots of preset color options.  Nice anti-aliasing.  Image can be downloaded.
Cons: Cannot specify image size (you need larger images for printing)

3. Julia and Mandelbrot Set Explorer

Pros: Allows you to set the image size in pixels.  Shows corresponding Julia set.
Cons: Navigation isn’t easy to use.  No anti-aliasing.

It reminded me of some of the patterns made by the cellular automata explored by Stephen Wolfram in his book A New Kind of Science (which can be viewed in its entirity here).  Automata are simple programs consisting of a row of cells that change color by following a few simple rules.  Many sets of rules produce repetitive or otherwise uninteresting patterns but some give rise to complex behavior.

But is this the only triangle dissection problem that leaves one unit unaccounted for? Can you invent another like this one?