1 Topic by William (edited by William 2022-12-13 23:39:32)

  • Registered: 2022-01-21
  • Posts: 1,198

Topic: An original random walk

I have devised a random walk rule that I hope might be original.

I have not looked at it in any detail.

I am wondering if it might be fun for readers here, including me, to explore together.

I wanted to devise a random walk that happens in a two-dimensional grid of squares, yet only involves flipping one coin.

So, please imagine a grid of squares, with one square labelled as the origin, and x axis and y axis as conventionally, and y very positive being North, so that we can have North, South, east, west and other directions.

There is one playing piece, that is circular and has an arrow drawn across a diameter.

At the start of a random walk the playing piece is in the origin square, with the arrow pointing along the x axis.

A coin is thrown.

If it lands with the head up, then the playing piece is rotated ninety degrees counterclockwise. If it lands with the tail up, then the playing piece is moved one square forward in the direction that the arrow is pointing, drawing a line as it moves showing where it has travelled.

Two basic thought experiment results are that in a unlimited number of throws of the coin if each throw of the coin results in "a head" then the playing piece never moves from the origin square yet rotates upon it. If each throw of the coin results in "a tail" then the playing piece moves along the x axis towards an infinite value of x.

I am wondering if the nature of the rule of the random walk means that some types of drawn path will be much more common than others.

William

  • Registered: 2022-01-21
  • Posts: 1,198

Re: An original random walk

This is becoming fascinating.

I did a thought experiment of considering every case of four flips of the coin.

I then did a thought experiment of what happens in a random walks with n throws if one result is a head all of the rest of them are tails, considering each case of where in the sequence of n results is the head.

I am now considering what happens if there are n throws of the coin and two results are a head and all the rest are tails.

I am trying to solve this by thought experiments, starting with n=2, then proceeding to n=3, n=4 and so on.

William

  • Registered: 2022-01-21
  • Posts: 382

Re: An original random walk

Here's a fascinating thought for you, William, in respect of

I did a thought experiment of considering every case of four flips of the coin

and

I then did a thought experiment of what happens......

I did a thought experiment of wondering if these thought experiments were actually analogue experiments.

(one) You presumably tabulated the results, and (two) your thoughts, or the entries, the data, in the table represented the real value of a flip, or whatever.

Discuss!!

  • Registered: 2022-01-21
  • Posts: 1,198

Re: An original random walk

Hello Jack, thank you for your interest.

Actually, I had not written it down. I was in bed, I had woken up early, and what with all the snow around and the cold weather, I decided to stay in bed for a while. So I carried out the thought experiments. smile

Then I posted about them in this thread after breakfast.

I approached the thought experiment by consider each of the possible sequences of heads, H and tails T, as if encoded as binary numbers with H=0 and T=1. This because of me being very familiar with four bit binary numbers.

I quickly realized that the direction in which the playing piece is pointing after the four flips of the coin have been acted upon depends solely on how many heads are in the sequence, yet the position of the playing piece after the four flips of the coin have been acted upon depends not only on the number of T results but as to where they are located in the sequence in relation to any H results in the sequence.

So I then considered the sequences in groups, with each sequence in a particular group having the same number of heads in the result.

This was interesting, as patterns emerge.

For example, with four flips of the coin, with one head in the sequence, there are four sequences. Each sequence results with the playing piece pointing North, with the playing pieces located in a diagonal line, at squares (0, 3), (1, 2) (2, 1), (3,0).

If one extends this to n flips of the coin, there are n possible sequences that each have one H and (n-1) T, and the n results are also a diagonal line from the y axis to the x axis.

Going back to four flips of the coin there are six possible sequences that have two and only two heads in the sequence. Pascal's triangle. These six sequences all end with the playing piece pointing West. The playing piece can end up at any one of six places, all in a pattern.

If there are three and only three heads in the four flips sequence then the playing piece ends up facing South, and there is a different pattern.

It is interesting how Pascal's triangle relates to this as the number, n, of flips increases, yet the rotation only has four possible results. For example, when n=4, the sequence of zero H results and the sequence of four H results each upon being acted upon result in the playing piece pointing East.

Yet for n flips, the sequence of zero H results and the sequence of n H results, upon being acted upon only result in the playing piece pointing in the same direction if n mod 4 = 0.

William

  • Registered: 2022-01-21
  • Posts: 1,198

Re: An original random walk

Thinking further about this, I am thinking that if a sequence of n flips starts with zero, one, two, or three H then has one and only one more H in it, then the set of all those sequences defines the outer limit of the results of any sequence of n flips, except for the n flip sequences in which there are no H at all after the zero, one, two or three at the start.

William