The collection

Every essay — page 27

Page 27 of 49, continuing through the fields in the same order.

Geometry Analysis Algebra Discrete Topology Probability Number Dynamics Logic Computation Applied What's new Series Concepts Search

Probability

Randomness with a shape.

One random path estimates the whole tree. probe columns 4,2,5,3,6,0; choices 8,5,3,1,2,1; estimate 769; exact 2057.

One random path down a search tree

How many matches a frustrating deal has. 0:0.016233/0.018316, 1:0.068899/0.073263, 2:0.144164/0.146525, 3:0.198194/0.195367, 4:0.201325/0.195367, 5:0.161112/0.156293, 6:0.105762/0.104196, 7:0.058554/0.059540, 8:0.027898/0.029770, 9:0.011615/0.013231, 10:0.004276/0.005292, 11:0.001406/0.001925, 12:0.000416/0.000642

A card game almost nobody wins

When to stop tossing a coin. 1:1 2:2 3:3 4:2 5:3 6:2 7:3 8:2 9:3 10:4 11:3 12:4 13:3 14:4 15:3 16:4 17:5 18:4 19:5 20:4 21:5 22:4 23:5 24:4 25:5 26:4 27:5 28:4 29:5 30:6

When to stop tossing a coin

The median has a bell of its own. normal: median sd 0.1238 (pred 0.1247), mean spread 0.0985; laplace: median sd 0.1071 (pred 0.0995), mean spread 0.1405; cauchy: median sd 0.1588 (pred 0.1563), mean spread 1.5006

The middle value has a bell of its own

Arcs that overlap and still leave gaps. n=14, a=0.2: starts 0.014,0.017,0.118,0.192,0.232,0.237,0.251,0.428,0.458,0.500,0.510,0.579,0.930,0.931; gaps 1; P(cover) exact 0.34675223.

Arcs that cover a circle

Two losing games with memory that win together. game A: -1.8000; game B: -1.9390; a fair coin chooses: 1.7434; the cycle A B B: 1.9077; ε=0.003.

Two losing games with a memory

How well a trimmed mean estimates the centre, shape by shape. Efficiency of the alpha-trimmed mean for normal: mean 1.000, 10% 0.943, 25% 0.837, median 0.637; logistic: mean 0.912, 10% 0.994, 25% 0.950, median 0.750; contam: mean 0.698, 10% 0.969, 25% 0.901, median 0.697; t5: mean 0.800, 10% 0.989, 25% 0.966, median 0.769; laplace: mean 0.500, 10% 0.669, 25% 0.815, median 1.000; t3: mean 0.500, 10% 0.932, 25% 0.984, median 0.811; cauchy: mean 0.000, 10% 0.419, 25% 0.785, median 0.811.

The best middle when the shape is unknown

What is left in the other box. Banach matchbox remainder distribution for N=50: means p=0.5: 7.0385, p=0.55: 9.9165, p=0.6: 16.1560; at p=0.5 P(0)=0.07959, P(1)=0.07959.

The matches left in the other pocket

Boxes as a permutation. A random permutation of 24 with cycle lengths 11, 7, 5, 1; longest 11.

A hundred prisoners and one shuffle

Three ways to draw a random chord. Samples of 220 chords from each of three procedures: random ends, random radius and random midpoint; chords longer than the triangle's side are coloured; chances 1/3, 1/2 and 1/4.

What a random chord is

One number for each arm. Gittins indices at discount 0.9 for s successes and f failures on a uniform prior: a fresh arm has mean 0.5 and index 0.7029.

One number for each arm

5 figures · new

A restaurant with room for one parameter. Seeded Chinese restaurants of 60 customers: 0.5: 4 tables, largest 44; 2: 10 tables, largest 17; 10: 22 tables, largest 10.

A restaurant and a shuffled deck

6 figures · new

Number

The whole numbers, and how much structure they turn out to have.

The sieve of Eratosthenes below 100. A grid of the whole numbers with the composites struck out by the prime that removes them.

The primes are what is left over

Euclid's construction on 2, 3, 5, 7. The product of the listed primes plus one, divided by each of them in turn; every division leaves one over.

There is no last prime

7 figures $φ as a continued fraction. The nested fraction, one quotient per step, descending to the right.$

A fraction that never closes

The Stern–Brocot tree to depth 4. Every positive rational, each appearing exactly once, generated by taking mediants.

Every fraction, exactly once

7 figures $8 multiples of φ in 7 boxes. The fractional parts of the first multiples of a number, dropped into equal boxes along the unit interval.$

How close a fraction can get

Two factor trees of 360. The same number split two different ways, both ending in the same primes.

One way to factor, and no other

The divisors of 60. Every divisor as a lattice point, one axis per prime, joined when one divides the other by a single prime.

The shape of a number's divisors

28 is perfect, because its divisors form this rectangle. Two rows of divisors: the powers of two, and the same powers multiplied by the Mersenne prime.

Numbers that are their own parts

The circle of radius √25 on the integer lattice. A circle drawn on the whole-number grid, with the lattice points it passes through marked.

Two squares, and a lattice

Rational points on the unit circle. Lines of rational slope through the left-hand point of a circle, each meeting it again at a rational point.

Every triple, on one circle

Necklaces of 5 beads in 2 colours. Every string of beads, grouped by the rotations that carry one onto another.

Necklaces that prove a theorem

One number, two dials: 3 and 5. A grid of remainder pairs, each cell holding the smallest number that leaves those two remainders.

Two dials at once