Grid and Probability

Consider the first quadrant of a 2D2D Cartesian system as a grid. A recursive function "marks" solely the boxes in the grid (one at a time) that neighbor exactly two other marked ones. Such boxes are referred to as "kink sites". The x and y axes are assumed to be already marked as an initial condition. At each iteration, all kink sites have equal probability of getting marked. So, e.g. starting with the unmarked grid the first and only box that can be marked is the one adjacent to the corner of the grid (box (1,1)(1,1) ). Subsequently, there are two options; either box (1,2) (1,2) above, or box (2,1) (2,1) to the right of the marked corner box, both equally likely to be marked with probability 0.50.5. The following kink sites keep getting marked in the same manner. The attached image (grid not shown) shows all marked boxes after 10,00010,000 computer simulated iterations. Different colors represent any bunch of 1,0001,000 iterations. As the number of marked boxes grows infinitely, if any, what algebraic function is traced by the kink site front?

Note by Tomasz Bukowy
3 months, 4 weeks ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link]( link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}


Sort by:

Top Newest

AIREp shares three domains with the Sp100 family, which are involved in the interaction of DNA-binding proteins and the regulation of transcription. The first is a homogeneously stained region (HSR domain) located between amino acids 1 and 106, which is expressed in all members of the Sp100 family and is thought to play a role in a variety of cellular processes, including growth control, apoptosis, and senescence. The second is the SAND domain located at amino acids 189-280. This domain is involved in chromatin remodeling activity.

Wendy Wilson - 2 months, 2 weeks ago

Log in to reply


Problem Loading...

Note Loading...

Set Loading...