% CHANGES TO VOLUME 1 OF THE ART OF COMPUTER PROGRAMMING
%
% Copyright (C) 2011, 2012, 2013, 2014, 2015, 2016, 2017 Donald E. Knuth
% This file may be freely copied provided that no modifications are made.
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\ \ifnum #1<97 \hundred#1\else19#1\fi}
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%%%%%%%%%%%%%% opening remarks %%%%%%%%%%%%%%%%%%%%%%%%
\def\lhead{INTRODUCTION}
\let\rhead=\lhead
\titlepage
\volheadline{THE ART OF COMPUTER PROGRAMMING}
\bigskip
\volheadline{ERRATA TO VOLUME 1 (after 2010)}
\bigskip
\noindent This document is a transcript of the notes that I have been making
in my personal copy of {\sl The Art of Computer Programming}, Volume~1 (third
edition, 27th printing), since it was first printed in 2011.
Previous errata are recorded in another file `{\tt all1-pre.ps}'.
\ifall Four levels of updates\dash---``errors,'' ``amendments,'' ``plans,''
and ``improvements''\dash---appear, indicated by four
\else Three levels of updates\dash---``errors,'' ``amendments,'' and
``plans''\dash---appear, indicated by three \fi
different typographic conventions:
\begingroup\def\hundred{17}
\bugonpage 0.666 line 1 (76.07.04)
Technical or typographical errors (aka bugs) are the most critical items, so
they are flagged with a `\thinspace{\manfnt x}\thinspace' preceding the page
number. The date on which I first was told about the bug is shown; this is the
effective date on which I paid the finder's fee. The necessary
corrections are indicated in
a straightforward way. If,~for example, the book says
`$n$' where it should have said `$n+1$', the change is shown thus:
\smallskip
$n$ \becomes $n+1$
\endchange
\amendpage 0.666 line 2 (89.07.14)
Amendments to the text appear in the same format as bugs, but without
the~`\thinspace{\manfnt x}\thinspace'. These are things I wish I had known
about or thought of when I wrote the original text, so I added them later.
The date is the date I drafted the new text.
\endchange
\def\hundred{19}
\planforpage 0.666 line 3 (17.11.20)
Plans for the future represent a third kind of item. In such notes I~sketched
my intentions about things that I wasn't ready to flesh out further when
I~wrote them down. You can identify these items because they're written in
slanted type, and preceded by a bunch of dots
`\hbox to 6em{\leaders\hbox to 5pt{\hss.\hss}\hfill}' leading to the date on
which I recorded the plan in my files.
\endchange
\improvepage 0.666 line 4 (38.01.10)
The fourth and final category\dash---indicated by page and line number in
smaller, slanted type\dash---consists of minor corrections or improvements
that most readers don't want to know about, because they are so trivial.
You wouldn't even
be seeing these items if you hadn't specifically chosen to print the complete
errata list in all its gory details.
Are you sure you wanted to do that?
\endchange
\endgroup
\ifall\else\medskip\ninepoint
My personal file of updates also includes a fourth category of items, not
shown in this list. They are miscellaneous minor corrections or improvements
that most readers don't want to know about, because they are so trivial.
If you really want to see all of the gory details,
you can download the full list from Internet webpage
$$\.{http://www-cs-faculty.stanford.edu/\char`\~knuth/taocp.html}$$
by selecting the ``long form'' of the errata.
\fi
\medskip
\tenpoint
My shelves at home are
bursting with preprints and reprints of significant research results that I
want to digest and summarize, where appropriate, in the ultimate edition of
Volume~1. I didn't do that in the third edition because I would surely have to do
it over again later: New results continue to pour forth at a great rate, and I
will have time to rewrite that volume only~once. Volumes 4 and~5 need to be
finished first. So I've put most of my effort so far into writing up
those parts of the total picture that seem to have converged to their
near-final form. It follows, somewhat paradoxically, that the updates in this
document are most current in the areas where there has been least activity.
On the other hand I do believe that the changes listed here bring Volume~1
completely up to date in two respects: (1)~All of the research problems in the
previous edition\dash---i.e., all exercises that were rated 46 and
above\dash---have received new ratings of 45 or less whenever I learned of a
solution; and in such cases, the answer now refers to that solution.
(2)~All of the historical information about pioneering developments has been
amended whenever new details have come to my attention.
\beginconstruction
The ultimate, glorious, 100\% perfect editions of Volumes 1--4A are works in progress.
Please let me know of any improvements that you think I ought to make.
Send your comments either by snail mail to D.~E. Knuth, Computer
Science, Gates Building 4B, Stanford University, Stanford CA~94305-9045,
or by email to
{\tt taocp{\char`\@}cs.stanford.edu}. (Use email for book suggestions
only, please\dash---all
other correspondence is returned unread to the sender, or discarded,
because I have no time to
read ordinary email.) Although I'm working full time on
Volume~4B these days, I~will try to reply to all such messages
within a year of receipt. Current news about {\sl The Art of Computer
Programming\/} is posted on
$$\.{http://www-cs-faculty.stanford.edu/\char`~knuth/taocp.html}$$
and updated regularly.
\par\endconstruction
\rightline{\dash---Don Knuth, January 2011}
\bigskip
\bigskip
{\quoteformat
What happened?
The subject took the bit in its teeth and ran away with it,
that's what happened.
I know now how Sir James Frazer felt when,
after setting out to dash off a brief monograph
on a single obscure rite, he found himself
in the embarrassing possession of
the 12 volumes of ``The Golden Bough.''
\author WAVERLEY ROOT (1974)
% International Herald Tribune, 22 May 1974, p8
\vfill\eject
}
\def\today{\number\day\space\ifcase\month\or
January\or February\or March\or April\or May\or June\or
July\or August\or September\or October\or November\or December\fi
\space\number\year}
%%%%%%%%%%%%%%% CHANGES FOR VOLUME 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\lhead{CHANGES TO VOLUME 1: FUNDAMENTAL ALGORITHMS}
\let\rhead=\lhead
\titlepage
\volheadline{FUNDAMENTAL ALGORITHMS}
\bigskip
\rightline{Copyright \copyright\ 2011, 2012, 2013, 2014, 2015
Addison\with Wesley; all rights reserved}
\rightline{Last updated \today}
\bigskip
\rightline{\sl Most of these corrections have already been made in
recent printings.}
\smallskip
\let\defaultpointsize=\tenpoint
\bugonpage 1.0 (on the back page of the dustcover, line 23) (12.09.10)
{\eightss Whatever your backgound, \becomes Whatever your background,}
\endchange
\bugonpage 1.x line 14 (16.03.24)
American Mathematical Monthly \becomes
The American Mathematical Monthly
\endchange
\amendpage 1.xi new paragraph to follow line 20 (13.11.25)
My efforts to extend and enhance these volumes have been enormously
enhanced since 1980 by the wise guidance of Addison--Wesley's editor
Peter Gordon. He has become not only my ``publishing partner'' but also
a close friend, while continually nudging me to move in fruitful directions.
Indeed, my interactions with dozens of Addison--Wesley people during more
than three decades have been much better than any author deserves. The
tireless support of managing editor John Fuller, whose meticulous
attention to detail has maintained the highest standards of production
quality in spite of frequent updates, has been particularly praiseworthy.
\endchange
\improvepage 1.xi line 5 before the author's signature (15.09.23)
nobody noticed \becomes nobody else noticed
\endchange
\amendpage 1.xvi line 16 (11.06.16)
a {\it 45\/} rating \becomes
a {\it 40\/} rating
\endchange
\amendpage 1.xvi line 20 (13.01.30)
creativity.\ \becomes
creativity. All exercises with ratings of {\it46\/} or more are open problems
for future research, rated according to the number of
different attacks that they've resisted so far.
\endchange
\improvepage 1.6 lines 1--3 (13.03.13)
$\{m,n\}$ \becomes $m$ and $n$ [twice]\nl
$\{n,r\}$ \becomes $n$ and $r$ [twice]
\endchange
\improvepage 1.8 lines $-5$ and $-4$ (13.03.13)
Such a computational method \dots\ it is also \becomes
Every step of such a computational method is clearly
effective, and experience shows that pattern-matching rules of this kind are
also
\endchange
\amendpage 1.9 the rating of exercise 7 (14.04.05)
\ninepoint {\it M21\/} \becomes {\it\HM21\/}
\endchange
\improvepage 1.11 lines 3--4 (15.05.06)
An expansion \dots\ can \becomes
A more leisurely presentation of much of the
following material, together with related concepts, can
\endchange
\bugonpage 1.17 line 26 (13.12.15)
{\bf11} (1838) \becomes {\bf12} (1838)
\endchange
\improvepage 1.18 clarification in exercise 4 (13.02.10)
\ninepoint $\phi^{n-2}$. \becomes $\phi^{n-2}$ for all positive integers $n$.
\endchange
\improvepage 1.19 clarification in exercise 5 line 1 (13.02.05)
exact divisors \becomes positive integer divisors
\endchange
\improvepage 1.22 lines 1 and 3 after {\eq(8)} (13.01.20)
$10^x$ \becomes $b^x$ \qquad[two places]
\endchange
\amendpage 1.23 line 1 after {\eq(13)} (15.02.22)
Edward M. Reingold. \becomes
Ernesto Trucco [{\sl Bull.\ Math.\ Biophysics\/ \bf18} (1956), 130].
\endchange
\bugonpage 1.27 line 1 (11.07.22)
\ninepoint $n>1$ \becomes $n>1$ and $n\ne e$.
\endchange % actually I've also changed $n$ to $x$ in this exercise
\bugonpage 1.50 line $-5$ (14.06.15)
{\sl della Scienze\/} \becomes {\sl delle Scienze\/}
\endchange
\amendpage 1.61 replacement for the bottom 7 lines (16.11.16)
\noindent
to use Eq.~\eq(20)\dash---which no longer applies.
In this situation
it pays to complicate things even {\it more}, by replacing the unwanted
$n+k\choose m+2k$ %$$\Bigl({n+k\atop m+2k}\Bigr)$$
by a sum of terms of the form
\smash{$x+k\choose2k$}, %$$\Bigl({x+k\atop 2k}\Bigr),$$
since our problem will then become a sum of problems that we know how to solve. Accordingly,
we use Eq.~\eq(25) with
$$k\mapsto j,\quad
r\mapsto n-1,\quad m\mapsto m-1,\quad s\mapsto k,\quad n\mapsto 2k;$$
all of the conditions required for \eq(25) are satisfied, because
$m$ and $n$ are positive.
\endchange
\amendpage 1.62 replacement for the top 18 lines (16.11.16)
The sum to be solved now takes the form
$$\sum_{k}\,\sum_{j=0}^{n-1}\Bigl({n-1-j\atop m-1}\Bigr)
\Bigl({k+j\atop2k}\Bigr)\Bigl({2k\atop k}\Bigr){(-1)^k\over k+1};\eqno(29)$$
and the rest is easy, by interchanging the order of summation:
Equation~\eq(28) tells us how to sum on~$k$, and
all the complications melt away. Everything reduces to just
$$\sum_{j=0}^{n-1}\Bigl({n-1-j\atop m-1}\Bigr)\,\delta_{j0},$$
so our final answer is
$$\Bigl({n-1\atop m-1}\Bigr).$$
The solution to this problem was fairly complicated, but not really mysterious;
there was a good reason for each step. The derivation should be studied
closely, because the use of \eq(25) illustrates some delicate maneuvering with
the conditions in our equations.
There is actually a better way to attack this problem, however! The reader
is invited to figure out how to transform the given sum so that
Eq.~\eq(26) applies (see exercise 30).
\endchange
\improvepage 1.70 in exercise 25 (13.10.07)
\ninepoint
line 1: as in Eq.~\eq(30). \becomes as in Example~4 (see Eq.~\eq(30)).\nl
line 5: the identity \becomes multiples of a special case of \eq(34),
\endchange
\bugonpage 1.70 line 2 of exercise 25 (13.10.07)
\ninepoint provided $z$ is small enough \becomes
provided that $x$ is close enough to 1
\endchange
\amendpage 1.79 new exercise for Section 1.2.7 (13.02.11)
\ninepoint
\ex25. [M21] Let $H_n^{(u,v)}=\sum_{1\le j\le k\le n}1/(j^uk^v)$.
What are $H_n^{(0,v)}$ and $H_n^{(u,0)}$?
Prove the general identity
$H_n^{(u,v)}+H_n^{(v,u)}=H_n^{(u)}H_n^{(v)}+H_n^{(u+v)}$.
\endchange
\amendpage 1.85 the displayed formula in exercise 24 (17.01.20)
insert `{\ninepoint det}' before the matrix
\endchange
\improvepage 1.104 line 11 (14.06.26)
random variable \becomes random integer
\endchange
\amendpage 1.104 new sentence preceding the exercises (12.07.19)
\noindent[S.~Bernstein had contributed key ideas in {\sl Uchenye zapiski
Nauchno-\nl\quad Issledovatel'skikh kafedr Ukrainy\/ \bf1} (1924), 38--48.]
\endchange
\amendpage 1.110 lines 14--16 (14.07.01)
We have \dots\ This equation \becomes We have
$$\root n\of{n}=
e^{\ln n/n}=1+(\ln n/n)+O\bigl((\log n/n)^2\bigr),\eqno(18)$$
because $\log n/n\to0$ as $n\to\infty$; see exercises 8 and~11.
(Notice that we needn't specify the base of `log' inside $O$-notation,
where constants are ignored.) This equation
\endchange
\amendpage 1.111 line above Fig.~12 (12.07.08)
{\sl Petropoli\-tan\ae\/} \becomes {\sl Imperialis Petropoli\-tan\ae\/}
\endchange
\amendpage 1.114 line 7 (13.02.26)
is less than \becomes is less in absolute value than
\endchange
\improvepage 1.116 line 2 of exercise 8 (12.02.18)
${cn^2\choose n}\big/c^n{n^2\choose n}$ \becomes
${cn^2\choose n}\big/\bigl(c^n{n^2\choose n}\bigr)$
\endchange
\amendpage 1.144 line 2 (11.11.29)
\ninepoint to zero. \becomes to zero,
and the overflow toggle is cleared.
\endchange
\amendpage 1.151 line $-11$ (11.11.29)
{\sl set to zero\/} \becomes {\sl set to positive zero\/}
\endchange
\improvepage 1.173 line 3 after the table (13.12.15)
table. \becomes
table, except that the unknown
destination of $e$ is represented there by `)' not `?'.
\endchange
\bugonpage 1.229 line 11 (11.02.06)
{\sl Mechanization\/} \becomes {\sl Mechanisation}
\endchange
\amendpage 1.229 lines 25--29 (15.12.19)
see {\sl Proceedings}\ \dots\ 87--90; B.~E. Carpenter \dots\ 78--79. \becomes
see B.~E. Carpenter \dots\ 78--79. See also H.~D. Huskey's mention of
``reversion storage'' in {\sl Proceedings}\ \dots\ 87--90.
\endchange
\amendpage 1.242 line 2 (12.04.28)
top, front \becomes top, bottom, front
\endchange
\bugonpage 1.268 program line {\it 71\/} (12.06.30)
\ninepoint \tt QLINK(F) \becomes QLINK[F]
\endchange
\improvepage 1.275 line 7 (12.09.06)
list have \becomes list must have
\endchange
\amendpage 1.303 in the line before {\eq(13)} (11.08.29)
transformation: \becomes
transformation (see M.~H. Doolittle,
{\sl Report of the Superintendent of the U.S. Coast and Geodetic Survey\/}
(1878), 115--120):
\endchange
\amendpage 1.331 rating of exercise 14 (12.08.15)
\ninepoint {\it22\/} \becomes {\it20\/}
\endchange
\bugonpage 1.339 replacement for line 17 (11.03.14)
$$D(y)=3\bigl( 1/(x+1)\bigr) -\bigl(-(a(2x))/(x^2)^2\bigr),\eqno(21)$$
\endchange
\amendpage 1.465 new quotation {(to follow the one by Lehmer)} (13.08.18)
{\quoteformat
I must explain, to begin with,
that all the Trees, in this system, grow {\rm head-downwards:}
the Root is at the {\rm top,} and the Branches are {\rm below.}
If it be objected that the name ``Tree'' is a misnomer, my answer
is that I am only following the example of all writers on {\rm Genealogy.}
A {\rm Genealogical$\!$} tree {\rm always} grows {\rm downwards:}
then why may not a {\rm $\,$Logical$\!$} ``Tree'' do likewise?
\author LEWIS CARROLL, in {\sl Symbolic Logic\/} (1896)
% book XII, chapter II; page 281 of Bartlett's editions
% galley proofs sent out 6 Nov 1896, then lost but eventually published 1977
}
\endchange
\let\defaultpointsize=\ninepoint % get ready for answer pages
\amendpage 1.466 append new copy at the end of answer 7 (14.08.10)
For example, try $m=5$ and $n=1$, 2, \dots: The average of
1, 2, 3, 2, 1,
3, 4, 5, 4, 2,
3, 4, 5, 4, 2, \dots\ is 3.6.
\endchange
\amendpage 1.467 new line at end of answer 8 (13.04.29)
\noindent
Each iteration either decreases $m$ or keeps $m$ unchanged and decreases~$n$.
\endchange
\amendpage 1.469 new lines at end of answer 8 (13.02.26)
\noindent
This construction can't make $d_k=9$ for all $k>l$, because that could
happen only if $(n+d_1/10+\cdots+d_l/10^l+1/10^l)^m\le u$.
\endchange
\improvepage 1.470 in step E1 of answer 28 (11.10.28)
If $1-\epsilon$ \dots $k\gets1$. \becomes
Set $x\gets1-\epsilon-x$, $y\gets y_0$,
and $k\gets1$, where $1-\epsilon$ is the largest possible value
of~$x$, and $y_0$ is the nearest approximation to $b^{1-\epsilon}$.
\endchange
\amendpage 1.472 new copy for end of answer 31 (11.10.20)
Consequently we have $\bigl(\sum_{j=1}^n u_j\bigr)\bigl(\sum_{j=1}^n v_j\bigr)
\le n\sum_{j=1}^n u_jv_j$ when $u_1\le u_2\le\cdots\le u_n$ and
$v_1\le v_2\le\cdots\le v_n$, a result known as {\it Chebyshev}'s
monotonic inequality. [See {\sl Soobshch.\ mat.\ obshch.\ Khar'kovskom
Univ.\ \bf4},\thinspace2 (1882), 93--98.]
\endchange
\amendpage 1.474 lines 3--5 of answer 43 (11.11.26)
as in exercise 44 \dots\ $(x_i-1)\bigr)$. \becomes
by setting $x=1$ in exercise~40 and obtaining
${\prod_{k\ne i}(x_k-1)/x_i\prod_{k\ne i}(x_k-x_i)}$. After multiplying
numerator and denominator by $x_i-1$, we can sum on $i$
by applying exercise~33 with $r=0$ to the $n+2$ numbers
$\{0,1,x_1,\ldots,x_n\}$.
\endchange
\amendpage 1.476 replacement for answer 4 (13.12.22)
\ans4. By part (f), $x\le\lceil x\rceil0$. The $k$th term is $r/(r-tk)$ times
$$\displaylines{\quad{1\over n!}\Bigl({n\atop k}\Bigr)
\prod_{0\le jn-1$ \becomes $y\ge n-1$
\endchange
\amendpage 1.491 replacement for last line of answer 67 (12.02.18)
${n\choose k}\ge\bigl({(n-k+1)e\over k}
\bigr){}^k{1\over ek}$, which is less memorable (but often sharper) than
${n\choose k}\ge\bigl({n\over k}\bigr){}^k$.
\endchange
\amendpage 1.492 replacement for answer 16 (16.06.21)
\ans16. $H_{2n}-\half H_n$.
\endchange
\amendpage 1.493 new answer for section 1.2.7 (13.02.11)
\ans25. $H_n^{(0,v)}=\sum_{k=1}^nH_n^{(v)}$ and $H_n^{(u,0)}=H_n^{(u-1)}$;
so the identity generalizes~\eq(8). [See L.~Euler,
{\sl Novi Comment.\ Acad.\ Sci.\ Pet.\ \bf 20} (1775), 140--186, \S2.]
\endchange
\bugonpage 1.503 last line of answer 3 (13.03.05)
1937 \becomes 1927
\endchange
\improvepage 1.504 last line of answer 8 (12.02.18)
${cn^2\choose n}\big/c^n{n^2\choose n}$ \becomes
${cn^2\choose n}\big/\bigl(c^n{n^2\choose n}\bigr)$
\endchange
\improvepage 1.514 replacement for line 5 of the program (13.04.30)
\nobreak\vskip-12pt\ansmixthree
3H \ \ \ &ENT2&9*8-8,1&Start at row 9.\cr
\endmix
\endchange
\bugonpage 1.514 replacement for lines 19 and 20 of the program (13.03.10)
\nobreak\vskip-12pt\ansmixthree
PHASE2&ENT3&9*8 \ \ \ &At this point $\rA =\min_jC(j)$\cr
3H&ENT2&0,3&Prepare to search a row.\cr
\endmix
\endchange
\amendpage 1.518 line 14 (14.08.27)
Victorius of Aquitania \becomes Victorius of Aquitaine
\endchange
\amendpage 1.546 last lines of answer 9 (14.06.30)
plagues many automatic programming systems.) \becomes
plagued many automatic programming systems in the old days.)
\endchange
\bugonpage 1.552 in answer 12 (12.06.30)
line 1: $29p$ \becomes $27p$\nl
line 3: $3\over4$ \becomes 78\%
\endchange
\amendpage 1.570 new sentence to follow line 1 of answer 30 (13.10.08)
Thus $\.{LOC(T)}=\.{HEAD}$, and $\.{HEAD}\symord/$ is the first node of
the binary tree in symmetric order.
\endchange
\improvepage 1.570 line $-3$ of answer 30 (13.10.08)
the algorithm of exercise 21 \becomes Algorithm~U in answer 21
\endchange
\amendpage 1.576 new answer(s) (14.09.09)
\anss 14, 15. See Martin Ward and Hussain~Zedan,
``Provably correct derivation of algorithms using FermaT,''
{\sl Formal Aspects of Computing\/ \bf26} (2014), 993--1031.
\endchange
\bugonpage 1.607 lines 1--9 (16.06.24)
[In some printings, the ``primes'' disappeared from step names;
for example, step C5$^\prime$ was simply called step C5.]
\endchange
\let\defaultpointsize=\tenpoint % begin appendices
\bugonpage 1.611 line 5 (14.07.25)
inner loop A3* \becomes inner loop A2.5*
\endchange
\bugonpage 1.613 last line of answer 28 (14.04.14)
$\.{AVAIL[$k$]}\gets \.L$. \becomes $\.{AVAILF[$k$]}\gets \.L$.
\endchange
\amendpage 1.624 through page 626 (12.04.07)
[replace the notation $(R\Relbar\joinrel\joinrel\joinrel\Rightarrow x;\;y)$
by $(R{?}\ x{:}\ y)$ in eleven places]
\endchange
\amendpage 1.628 new entries for Appendix C (11.04.17)
\ninepoint\noindent
Program 1.2.10M, 145, 186.\nl
Program 1.4.3.1M, 204--211, 530.\nl
Program 2.1A, 236.\nl
Program 2.1B, 535.\nl
Program 2.3.1S, 325.
\endchange
\bugonpage 1.629 in Appendix C (11.04.17)
\ninepoint\noindent
Algorithm 2.4B$'$, 606. \becomes
Algorithm 2.4B$'$, 605. Algorithm 2.4B$''$, 606.\nl
Algorithm 2.5G, 613. \becomes Algorithm 2.5G, 613--614.
\endchange
\expandafter\ifx\csname indexeject\endcsname\relax\else\vfill\eject\fi
\amendpage 1.630 and following (11.01.01)
Miscellaneous changes to the existing index of Volume~1 are collected here,
including corrections and amendments to the old entries as well as new entries
that are occasioned by the new material. Thus, the lines of the full index
that have changed serve also as an index to the present document. However,
when a correction or amendment has caused an old index entry to be deleted,
the deletion is usually not indicated.
\input exotic
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\hangindent 2em
Bernstein, Sergei Natanovich ({\rus Bernshtei0n, Sergei0 Natanovich}), 104. %30th
Carroll, Lewis (= Dodgson, Charles Lutwidge), 465. % 32nd
Chebyshev (= Tschebyscheff), Pafnutii Lvovich ({\rus Chebyshevp2, Pafnut{\char12}i0 Lp1vovichp2} = {\rus Chebyshev, Pafnutii0 Lp1vovich}), inequalities, 98, 104, 472. % 31st
Comfort, Webb Tyler, 461. % 34th
Contour integration, 49, 92. % 37th
Depth of node in a tree, \see Level. % 30th
Diagrams of structural information, tree structures, 309--315, 337, 346, 349, 460, 465. % 32nd
Dodgson, Charles Lutwidge, \see Carroll. % 32nd
Doolittle, Myrick Hascall, 303. % 29th
$e$ (base of natural logarithms), 23, 619--620, 626. % 30th
Empty list, 244--245, 247, 258, 260--261, 273--275, 278, 280, 540, 546. % 30th
Euler, Leonhard ({\rus Ei0lerp2, Leonardp2} = {\rus E1i0ler, Leonard}), 49, 50, 52, 57, 75, 76, 87, 111, 374, 407, 472, 493, 496, 536, 600. % 31st
Family trees, 310--311, 317, 406, 465. % 32nd
Fuller, John Edward, xi. % 32nd
Geek art, xx. % 34th
GO button of \MIX, 126, 139, 143--144, 211. % 29th
Gordon, Peter Stuart, xi. % 32nd
Huskey, Harry Douglas, 229. % 34th
Knuth, Nancy Jill Carter (\GC75:75:-3:62% G2463
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), x, xx. % 33rd
K\"onigsberg bridges, 374. % 36th
Lam\'e, Gabriel L\'eon Jean Baptiste, 407. % 34th
Lilius, Aloysius (= Aloigi Giglio, Luigi Lilio), 159. % 33rd
Lineal chart, 310--311, 465. % 32nd
log (without subscript), 110. % 33rd
Markov, Andrei Andreevich ({\rus Markov, Andrei0 Andreevich}), the younger, 9. % 31st
Mealy, George Henry, II, 462. % 36th
Neumann, John von (= Neumann J\'anos Lajos = Margittai Neumann J\'anos), 18, 229, 231, 457. % 34th
Nonnegative coefficients, 396, 501. % 30th
Number system, decimal, 21, 619. % 32nd
Overflow toggle of \MIX, 126, 131, 134, 142, 144, 208, 214, 228. % 29th
Pattern matching, 8. % 31st
Pedigree, 310--312, 465. % 32nd
Reingold, Edward Martin ({\heb \Hdd/\Hll/\Hvv/\Hgg/\Hnn/\Hyy/\Hyy/\Hrr/},\indexbreak{\heb \Hfm/\Hyy/\Hyy/\Hkh/ \Hfn/\Hbb/ \Hhh/\Hsh/\Hmm/ \Hqq/\Hkh/\Hts/\Hyy/}), 518. % 33rd
Residue theorem, 472--473. % 37th
Reversion storage, 229, 240. % 34th
Root of a tree, 308, 309, 317, 465. % 32nd
Staver, Tor B{\o}hm, 582. % 28th
Stirling numbers, 66--69, 71--74, 78, 99--100, 506, 582, 591, 625. % 34th
Trees, diagrams of, 309--315, 337, 346, 349, 460, 465. % 32nd
Trucco, Ernesto, 23. % 33rd
Victorius of Aquitaine, 518. % 33rd
von Neumann, John (= Neumann J\'anos Lajos = Margittai Neumann J\'anos), 18, 229, 231, 457. % 34th
Ward, Martin Paul, 576. % 32nd
Windley, Peter Francis, 523. % 34th
Zeckendorf, \smash{\'E}douard, 495. % 36th
Zedan, Hussein Saleh Mamoud\indexbreak
({\arab\An/\Aa/\Afd/\Aiy/\Az/ \Ad/\Afw/\Amm/\Amhh/\Aim/ \Afhh/\Ail/\Afa/\Aiss/ \Afn/\Amy/\Ams/\Aihh/}), 576. %32nd
\vfill
\enddoublecolumns
\endchange
\bye
[The next printing will be the 37th.]
Not mentioned above:
ARTICLES "TO APPEAR" THAT ARE STILL PENDING: