The third curve shown on page 346 is obtained from Table[Cost[IntegerDigits[i, 2, n]], {i, 0, 2 n - 1}] There is no single ordering that makes all states which can be reached by changing a single square be adjacent. However, the ordering defined by GrayCode from page 901 does do this for one particular sequence of single square changes.

However, with the rule n  If[EvenQ[n], 3n/2, Round[3n/4]] it is always possible to go backwards by the rule n  If[Mod[n,3]  0, 2n/3, Round[4n/3]] The picture shows the number of base 10 digits in numbers obtained by backward and forward evolution from n = 8 . … But apart from these cycles, the numbers produced always seem to grow without bound at an average rate of 3/(2 √ 2 ) in the forward direction, and 2 4 1/3 /3 in the backward direction (at least all numbers up to 10,000 grow to above 10 100 ). Approximately one number in 20 has the property that evolution either backward or forward from it never leads to a smaller number.

Evaluation chains The idea of building up computations like 1 + 1 + 1 + … from partial results has existed since Egyptian times. … The method based on IntegerDigits in the previous two notes can be improved (notably by power tree methods), but apparently about Log[t] steps are always needed.

If a large amount of numerical data has been made up by a person this can be detectable through statistical deviations from expected randomness—particularly in structural details such as frequency of digits.

The following definition also handles the more general case of r neighbors: CAStep[TotalisticCARule[rule_List, r_Integer], a_List] := rule 〚 -1 - Sum[RotateLeft[a, i], {i, -r, r}] 〛 One can generate the representation of totalistic rules used by these functions from code numbers using ToTotalisticCARule[num_Integer, k_Integer, r_Integer] := TotalisticCARule[IntegerDigits[num, k, 1 + (k - 1)(2r + 1)], r]

Note (e) for Randomness from the Environment…An early example was the ERNIE machine from 1957 for British national lottery (premium bond) drawings, which worked by sampling shot noise from neon discharge tubes—and perhaps because it extracted only a few digits per second no deviations from randomness in its output were found. … All sorts of schemes have been invented for getting unbiased output from such systems, and acceptable randomness can often be obtained at kilohertz rates, but obvious correlations almost always appear at higher rates. … It still seems likely however that some general inequalities should exist between the rate and quality of randomness that can be extracted from a system with particular thermodynamic properties.

Mathematica is available from Wolfram Research for all standard computer systems; much more information about it can be found on the web, especially from www.wolfram.com . … Here are examples of how some of the basic Mathematica constructs used in the notes in this book work: • Iteration Nest[f, x, 3] ⟶ f[f[f[x]]] NestList[f, x, 3] ⟶ {x, f[x], f[f[x]], f[f[f[x]]]} Fold[f, x, {1, 2}] ⟶ f[f[x, 1], 2] FoldList[f, x, {1, 2}] ⟶ {x, f[x, 1], f[f[x, 1], 2]} • Functional operations Function[x, x + k][a] ⟶ a + k (# + k&)[a] ⟶ a + k (r[#1] + s[#2]&)[a, b] ⟶ r[a] + s[b] Map[f, {a, b, c}] ⟶ {f[a], f[b], f[c]} Apply[f, {a, b, c}] ⟶ f[a, b, c] Select[{1, 2, 3, 4, 5}, EvenQ] ⟶ {2, 4} MapIndexed[f, {a, b, c}] ⟶ {f[a, {1}], f[b, {2}], f[c, {3}]} • List manipulation {a, b, c, d} 〚 3 〛 ⟶ c {a, b, c, d} 〚 {2, 4, 3, 2} 〛 ⟶ {b, d, c, b} Take[{a, b, c, d, e}, 2] ⟶ {a, b} Drop[{a, b, c, d, e}, -2] ⟶ {a, b, c} Rest[{a, b, c, d}] ⟶ {b, c, d} ReplacePart[{a, b, c, d}, x, 3] ⟶ {a, b, x, d} Length[{a, b, c}] ⟶ 3 Range[5] ⟶ {1, 2, 3, 4, 5} Table[f[i], {i, 4}] ⟶ {f[1], f[2], f[3], f[4]} Table[f[i, j], {i, 2}, {j, 3}] ⟶ {{f[1, 1], f[1, 2], f[1, 3]}, {f[2, 1], f[2, 2], f[2, 3]}} Array[f, {2, 2}] ⟶ {{f[1, 1], f[1, 2]}, {f[2, 1], f[2, 2]}} Flatten[{{a, b}, {c}, {d, e}}] ⟶ {a, b, c, d, e} Flatten[{{a, {b, c}}, {{d}, e}}, 1] ⟶ {a, {b, c}, {d}, e} Partition[{a, b, c, d}, 2, 1] ⟶ {{a, b}, {b, c}, {c, d}} Split[{a, a, a, b, b, a, a}] ⟶ {{a, a, a}, {b, b}, {a, a}} ListConvolve[{a, b}, {1, 2, 3, 4, 5}] ⟶ {2a + b, 3a + 2b, 4a + 3b, 5a + 4b} Position[{a, b, c, a, a}, a] ⟶ {{1}, {4}, {5}} RotateLeft[{a, b, c, d, e}, 2] ⟶ {c, d, e, a, b} Join[{a, b, c}, {d, b}] ⟶ {a, b, c, d, b} Union[{a, a, c, b, b}] ⟶ {a, b, c} • Transformation rules {a, b, c, d} /. b  p ⟶ {a, p, c, d} {f[a], f[b], f[c]} /. f[a]  p ⟶ {p, f[b], f[c]} {f[a], f[b], f[c]} /. f[x_]  p[x] ⟶ {p[a], p[b], p[c]} {f[1], f[b], f[2]} /. f[x_Integer]  p[x] ⟶ {p[1], f[b], p[2]} {f[1, 2], f[3], f[4, 5]} /. f[x_, y_]  x + y ⟶ {3, f[3], 9} {f[1], g[2], f[2], g[3]} /. f[1] | g[_]  p ⟶ {p, p, f[2], p} • Numerical functions Quotient[207, 10] ⟶ 20 Mod[207, 10] ⟶ 7 Floor[1.45] ⟶ 1 Ceiling[1.45] ⟶ 2 IntegerDigits[13, 2] ⟶ {1, 1, 0, 1} IntegerDigits[13, 2, 6] ⟶ {0, 0, 1, 1, 0, 1} DigitCount[13, 2, 1] ⟶ 3 FromDigits[{1, 1, 0, 1}, 2] ⟶ 13 The Mathematica programs in these notes are formatted in Mathematica StandardForm .

This growth can be viewed as a consequence of potentially having to propagate carry digits from one end of the input number to the other.

Blocks in such sequences obtained from Partition[list, n, 1] must all be distinct since they correspond to successive complete states of the shift register. … (Related sequences can be generated from RealDigits[1/p, 2] as discussed on page 912 .)

And there is certainly in general no lack of radio signals that we receive from around our galaxy and beyond. … But could it be that some of these signals instead come from extraterrestrial intelligence—and are in fact meaningful messages? … And although I somewhat doubt it, one could certainly imagine that if one were to show data like the center column of rule 30 or the digit sequence of π to an extraterrestrial then they would immediately be able to deduce simple rules that can produce these.