Nine walks, and the square root
walk is one function. Everything below came out of it during this
build, at parameters taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and if the generator changes, this page
changes with it.
With nothing chosen
Where the walks ended
The same walk, on a grid
Coming home, in one, two and three dimensions
Walks that prefer the steps they have taken
Every count of reds equally likely
What it checks while it draws
Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.
- the enumeration finds the known number of 0-step walks ×20
- the measured exponent is the larger of ½ and ρ (a = 0) ×9
- the formula agrees with the enumeration at 0 steps above ×7
- the sum of 4 steps has probabilities adding to one ×4
- the sum of 4 steps has some spread to speak of ×4
- the share of red after 1000 draws follows the beta law with parameters 1 and 1 ×3
- between 2 and 5 steps ×2
- in 2 dimensions the measured range is within 25% of the classical law at 10,000 steps ×2
- 10 to 200 matches a box ×1
- A and B each lose over 300 rounds and the random mixture wins ×1
- A B B is the best of the sixteen ×1
- a bell below the threshold, two humps at ρ = 3/4, uniform for Pólya ×1
- a walk in space escapes, and the simulation sees it ×1
- a walk on a line comes back, and the horizon barely hides it ×1
- a walk on a plane comes back too, but more slowly ×1
- a walk takes a whole positive number of steps ×1
- above it like n to the power ρ (Pólya, ρ = 1) ×1
- above it like n to the power ρ (ρ = 3/4) ×1
- above one half it stays well away from nought ×1
- an endpoint has the parity of the step count ×1
- an even number of steps between 8 and 200 ×1
- and impossible with everything ×1
- and is then exactly fair ×1
- and moves the start to its mirror image ×1
- and so are the reflected ones ×1
- and the distribution is symmetric ×1
- and the gap to the bell curve closes at every step ×1
- and the variance of its position is exactly the number of steps ×1
- and their spread is the square root of the number of steps ×1
- at the threshold slightly faster than √n, by a logarithm ×1
- at ε = 0 game B spends exactly 5/13 of its time on the bad coin ×1
- at ρ = 1/3 every start heads for the bell ×1
- at ρ = 3/4 the shape is set by the start and barely moves with n ×1
- B's capital avoids remainder 1, and mixing widens the distribution ×1
- below one half the excess kurtosis dies away ×1
- below the threshold the spread grows like √n (red adds a blue) ×1
- below the threshold the spread grows like √n (ρ = 1/3) ×1
- between 4 and 16 steps are drawn, since every path of them is enumerated ×1
- dim is read only by the range view ×1
- each exact distribution is of between 2 and 400 steps ×1
- each game alone loses ×1
- each step moves one place along one axis ×1
- every game drawn ends at a barrier ×1
- every order of the same draws has the same chance ×1
- every path from below the level has to cross it ×1
- every path is counted once ×1
- every step is drawn ×1
- every step is one place, either way ×1
- every step is to a neighbour ×1
- every walk is counted once ×1
- folding leaves the endpoint alone ×1
- folding matches the touching paths with the paths from the mirrored start one for one ×1
- from one of each, every count of reds has chance exactly 1/(n + 1) ×1
- in 1 dimension the measured range is within 6% of the classical law at 10,000 steps ×1
- in one dimension the range is the width of the explored interval ×1
- lattice is read only by the sawcount view ×1
- mixing the two losing games wins ×1
- most urn walks make their last return to the start within the first hundred steps ×1
- playing only B or only A loses and some mixture wins ×1
- ruin is certain with nothing left ×1
- simulated urns agree with the exact law ×1
- some coin settings give the paradox ×1
- some walk goes outside the envelope, because √n is typical and not maximal ×1
- steps is not read by the views that fix their own lengths ×1
- ten draws decide most of the spread above one half and little of it below ×1
- the chances add to one ×1
- the distribution sums to one ×1
- the drawn path really does touch the level ×1
- the drawn walk is in one or two dimensions ×1
- the drawn walk's mean square displacement per step is near one in every window ×1
- the drawn walks are exactly the known number ×1
- the durations satisfy their own recurrence ×1
- the enumerated check runs at an even length between 4 and 16 ×1
- the enumeration finds the known counts ×1
- the extremes are more likely than the middle, which is the whole surprise ×1
- the full distributions agree with the residue chain's expected capital ×1
- the lattice is the square grid or the honeycomb ×1
- the least likely outcome is an even split ×1
- the mixture wins at ε = 0 and stops winning at some handicap below 0.02 ×1
- the mixture wins over a wide range of γ ×1
- the one-dimensional survival matches C(2m, m)/4^m ×1
- the option is one this view reads ×1
- the paths ending where they end are counted by a binomial coefficient ×1
- the player starts with something and not everything ×1
- the position has the parity of the step count ×1
- the probabilities of every remainder add to one ×1
- the quoted constant is Pólya's ×1
- the random mixture wins for every ε below about 0.0131 ×1
- the range is the number of distinct sites on the path ×1
- the ruin probabilities satisfy their own recurrence ×1
- the sampled walk never revisits a site ×1
- the share decided grows with m and stays between nought and one ×1
- the spread of the endpoints is of the order of the square root of the steps ×1
- the squared distance grows faster than the simple walk's n, with exponent near 3/2 ×1
- the table holds between 4 and 40 units ×1
- the three windows are whole numbers of steps ×1
- the three windows are whole numbers of steps inside the walk ×1
- the time spent above is always even ×1
- the urn is balanced: every draw adds the same number of balls ×1
- the urn's expected returns are half the logarithm of n plus a constant ×1
- the view is one the family draws ×1
- the walk can reach its endpoint in the steps allowed ×1
- the walk comes back to where it started at least once ×1
- the walk has no drift ×1
- the walk is between 400 and 60,000 steps long ×1
- the walk starts above the level it may touch ×1
- the walks are centred on where they started ×1
- the walks leave at very different speeds ×1
- the win probability is a probability ×1
- three-dimensional walks that have not returned by step 2,000 are about 1 − 0.3405 of them ×1
- with no handicap game B is exactly fair ×1
Where it is called
Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.
A walk that always comes home, until it does not
Step left or right at random, forever, and the walk returns to where it started with certainty. On a grid it also returns. In space it does not, and about a third of walks leave and never come back.
ProbabilityA walk that follows its own footsteps
Let a walk step up with probability equal to the share of up-steps it has taken so far, and every count of up-steps after twenty steps is exactly equally likely. The walk leaves along a straight line at a speed chosen by its first few steps, and although it is expected to come back to its start infinitely often, almost every run comes back only a few times.
ProbabilityA walk that may not step where it has been
Forbid a walk on the square grid from ever revisiting a site and the number of possible n-step walks grows like 2.638ⁿ instead of 4ⁿ — a number nobody can write down exactly. On the honeycomb it is exactly √(2 + √2), proved in 2010. And the walks spread out like n to the three-quarters, faster than any ordinary walk, which physicists have used since 1949 and mathematicians still cannot prove.
ProbabilityA walk that samples a distribution
When a distribution can be evaluated but not drawn from, a wandering point can be arranged to visit each state as often as its weight says. The rule needs no normalising constant, compares two weights and steps or stays.
ProbabilityAn average that never settles
The average of many independent quantities is supposed to steady as their number grows. For one famous distribution it does not steady at all — the average of a thousand draws has exactly the same distribution as a single draw, and no amount of further averaging changes it.
ProbabilityAn urn forgets its start only below one half
Let each draw from an urn add balls of both colours in fixed amounts, and the long run depends on a single ratio of two eigenvalues. Below one half the urn behaves like a coin, its fluctuations spread like the square root of the draws and settle into a bell. Above one half the first few draws decide most of the outcome, the spread grows faster, and the shape that results is not a bell and depends on how the urn began.
ProbabilityHalf the time is the rarest answer
In a fair game of many rounds, the fraction of the time one side is ahead is not usually near a half. It is usually near nought or one, and an even split is the single least likely outcome there is.
ProbabilityHow far from the average a thing can be
Knowing only an average and a spread — nothing about the shape, nothing about the number of outcomes, nothing about symmetry — the chance of landing three standard deviations out is at most one in nine. And there is a distribution that lands there exactly that often, so the bound cannot be improved.
ProbabilityHow fast the bell arrives
The limit theorem says a standardised sum approaches the bell curve and says nothing about when. The rate is one over the square root of the number of terms, the constant in front is made of the third moment, and both are visible.
ProbabilityThe chain that stops
Give a chain a state it cannot leave and there is no long run to find — every walk ends. What is worth computing instead is how long it lasts and where it finishes, and both are exact answers to a linear system rather than limits of anything.
ProbabilityThe ground a walk covers
A random walk of a thousand steps visits far fewer than a thousand places: in one dimension about fifty, in the plane about four hundred, in space about six hundred and sixty. The share of steps that land on new ground is exactly the chance of never coming home — so the number that decides whether a walker returns also decides how much of the world it sees.
ProbabilityThe matches left in the other pocket
A smoker keeps a box of fifty matches in each pocket and picks a pocket at random. When one box is found empty, the other holds about seven, and none and one are exactly equally likely. Favour one pocket by a tenth and the answer is sixteen; give each box five thousand matches and a fair choice leaves seventy-nine while a bias of one in a hundred leaves nearly two hundred — because a fair remainder grows like a square root and any bias makes it grow in proportion, with the change-over at a bias of one over the square root of the box.
ProbabilityThe path folded at its first touch
Counting the walks that touch a line looks like a question about a walk's whole history. Fold each one where it first touches, and it becomes a question about where walks end up — which is a binomial coefficient, and is already known.
ProbabilityThe time spent and the share held
Stationary shares are a limit of distributions — where the walk probably is after many steps. Here the question is about a single walk: the fraction of its time spent in each state is that state's share, and the expected wait between visits is exactly the reciprocal.
ProbabilityThe walk that becomes a curve
Shrink the steps of a random walk and it disappears. Shrink them while stretching the time in the right proportion — space by the square root of whatever time is divided by — and something is left behind, which is a curve nobody could draw.
ProbabilityTwo barriers and a fair game
A fair walk between two absorbing barriers is ruined with a probability that is a straight line in the starting stake, and lasts for a number of steps that is the product of what each side can lose. Both facts come from the same two-line recurrence, and both are bad news for the smaller player.
ProbabilityTwo losing games that win together
Game A is a coin that wins 49.5% of the time. Game B tosses a bad coin when the capital is a multiple of three and a good one otherwise, and it loses too, because the capital spends more than a third of its time on multiples of three. Choose between the two games at random and the walk drifts upwards by 0.0157 a round; play A, B, B over and over and it gains 0.0574. Nothing is wrong with the arithmetic. The losing coin A wins by knocking the capital off the bad remainder.
ProbabilityTwo losing games with a memory
Parrondo's first paradox needed a game that watched the player's capital. This one watches nothing but the last two results: a coin that wins nine times in ten after two losses, one time in four after a loss and a win, seven times in ten after two wins. Alone it loses, and so does a slightly unfair coin; played at random together they win. The walk treats every position alike, and the paradox survives — because it was never about position.
ProbabilityWhere the shares have nowhere to go
On finitely many states, a chain that can reach everywhere and is not forced into a rhythm settles down. Give it infinitely many and both conditions can hold while the walk leaves and never returns — or returns with certainty and takes an unbounded average time about it.