1. HexCellEnt: A 2-D Random Walk Game
Game Basics: Place your marker in the center hexagon; roll a die; move one space in the direction indicated by the 6-point rose. Repeat.
Goal: Move your marker off the game board (minimum number is 4 rolls).
Questions: What is the probability of exiting the board in exactly 4 rolls? How could you calculate this probability?
Ask students if any were able to calculate the probability {write down names of any getting the correct answer, for extra credit}. If so, ask them how they got the answer.

2. Brute-force Method: A Tree Diagram
Point out the two cells with two lines leading in to them (i.e gets you to the same cell as 1-6-1). Note that those paths out count double!
Result: 15 x 6 = 90 different 4-step escape paths, out of 64 = 1296 different possible 4-step paths…
P(out in 4 rolls) = 90/1296 ≈ {approx. 6.94% chance of exiting in 4 rolls}
That’s swell… but what about P(out in 5 rolls)… or P(out in “n” rolls)??

3. A Better way to Count Paths1 6 2
Have students calculate the 3rd roll by hand… or at least the outer ring (that way they wouldn’t need a game board template). Also ask, “What is the total of all the numbers on the board?” {Ask for each board}.
After Roll #1
After Roll #2

4. Can you describe the patterns?12 15 6 1 3 2
After Roll #2
After Roll #3

5. After Roll #3 After Roll #4
90 paths escape in 4 rolls: P(escape in 4) = 90/1296 ≈
Repeat the question from slide 2 {what about the probabilities: P(out in 5 rolls) and P(out in “n” rolls)?}.
After Roll #3
After Roll #4

6. 3-ring HexCellEnt board through 30 rolls; Ptot ≈ 95.0%
Once I set up the Excel file to calculate the numbers in each cell, it’s easy enough to calculate the probability of exiting after any specific number of rolls. Here, I’ve gone up to 30 rolls, by which point, about 95% of the games are finished. The Excel calculations show other interesting things, like that about 50% of all games should end in 10 or fewer rolls. Our record this year was 60 rolls. According to the Excel calculations, only about 2 games out of 1,000 should take that long!
“How can I be so confident in these facts? {Answer: even though each move is random, so each game has a random outcome, the results are predictable because this graph essentially represents infinitely many games!}

7. Now, after the first 100 games played by students (first time I did this activity), the agreement between observed and expected was not so great! (E.g. there were way too many 8s, and way too few 14s and 19s).

8. After 200 games, the 8s and 19s have settled down, but the 14s are still too low.

9. Do you see where this is headed?
Do you see the general trend here?

10. Although the outcome of each game is random…
The bars essentially represent infinitely many games played (though we’d have to change “frequency” to the respective “probability”). Now, infinity is a funny number (it’s really, really, really big). Realistically, how many games do you think we’d need to play in order to get the “observed” to line up with the “expected”?
The histogram bars represent the playing of how many games?

11. “Occupation Numbers” Revisited
Note that after the 2nd roll there are 36 possible 2-step paths. The numbers in the figure show how many of these lead to each cell. If you divide these numbers by “36”, you’ll get the fraction of 2-step paths leading to each cell. In fact, this is also the probability of reaching each cell in 2 rolls, assuming you play a very large number of games!

12. “Occupation Numbers” Revisited
This is where a big conceptual shift occurs, so I should leave this slide up (and tell students to think and talk this over with neighbors) while I distribute the handout with these figures {for rolls 2 – 4}. Also, before I leave the front of the classroom, I spray a little cologne on the front table. When I return to the front, ask if there are questions about relative concentration [one biggie is that they add to “1” until the particles start leaving the board]. Hopefully, some of the students have by now noticed the cologne scent and have asked about it (see next slides).
Occupation probabilities, OR…
Relative Concentrations
Occupation numbers

13. HexCellEnt: It’s not just a game… it’s a simulation… of diffusion!
First task: Determine how far, on average, a particle travels per unit time (i.e. roll number). Fraction that have gone 0 steps: (6/36) x 0 = 0 Fraction that have gone 1 step: (12/36) x 1 = … Fraction that have gone 2 steps: (18/36) x 2 = … Now add the terms! This is a “weighted average”. Use slide #4 above to calculate the average distance traveled after roll #3.

14. A “Famous” Diffusion Law Result
The graph at right shows the results of the weighted averages you just calculated, but this time for a much larger game board.
The equation for the best-fit curve is consistent with a well-known result for a random walk – a power law relationship between distance traveled and number of steps taken. In words, you would say: “The distance traveled is proportional to the ______ of the number of steps.”

15. Advanced Concept: Difference in Concentrations
Here, we’re focusing on the border between the 1st and 2nd rings (boldface). For every side along this border, calculate the difference of the two numbers across the border, add up the total of these, and then divide by 6n {where n = roll number}. Do this for roll numbers 2 – 5.

16. Advanced Concept: The Concept of Flux
Flux is a “flow rate”: the rate at which something flows through an imaginary unit of surface area. Units: (# of particles/s)/cm2

17. Advanced Concept: Flow Rate across a Border
Outbound flux: Notice that all cells in ring #1 have three sides exiting to ring #2. For each cell: Fluxout = (2 * 3/6) ÷ 36 Inbound flux: Some cells have 1 side leading in and some have 2 sides. Total inbound flux is: Fluxin = [(1*6*1/6)+(2*6*2/6)] ÷ 36 Do this for roll numbers 2 – 5.

18. Alternatively – for the Inquisitive Student
Slides above showed some advanced concepts for students interested in pursuing these ideas. Hopefully you’ve discovered some patterns, so you might consider pursuing alternative activities on your own. I have created Excel files to study the following alternatives:
Larger hexagon boards – up to 17 concentric rings
Other plane tiling arrangements (e.g. a “great rhombitrihexagonal” tiling: dodecagon, hexagon & square)
Soccer-ball-shaped HexCellEnt board
A 3-D space-filling shape – the truncated octahedron