[tt] NS: John Lucas: Necessary Evil

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John Lucas: Necessary Evil
http://www.amconmag.com/2008/2008_05_19/article2.html
8.6.2

Churchill, Hitler, and the Unneces-sary War: How Britain Lost Its
Empire and the West Lost the World, Patrick J. Buchanan, Crown, 501
pages


Patrick Buchanan's new book contains two themes under one cover. One
is addressed to the present, the other to the Second World War. One
is his declaration that the American empire is in great and deep
trouble--that, like the British Empire two thirds of a century ago,
it is overextended and weak. The other is that the Second World War
was a grievous mistake--that Britain (foremost: Churchill) and
America should not have fought Hitler's Germany. The two themes are
not equivalent, and their treatment in this book is uneven. The vast
majority of pages are about World War II. But in Buchanan's mind the
two themes are obviously inextricable, indeed, dependent on each
other. For the purpose of a review, however, I must separate them.

That the present American empire is much overextended, overgrown,
and at risk of all kinds of dangers, most of them willfully ignored
by the American people and their politicians, is so. Buchanan
deserves credit for having pointed this out, again and again, in his
articles and books. But, alas, in his discussion of his larger
thesis, his arguments are stamped by what we might call selective
indignation or, more accurately, special pleading. (Indignation,
after all, is almost always selective, while not every pleading is
necessarily special.)

He claims that the transformation of the United States from a
Republic to an empire was started by George W. Bush. What Bush has
done and is still doing is, of course, lamentable. But the reaching
out of American power all over the world, the fact that there are
now American bases and missions in more than 700 places around the
globe, the building of a 600-ship Navy, etc., began with Eisenhower
and Dulles. It went on with Johnson, Nixon, Carter, and especially
with Buchanan's hero, Reagan, and then under Clinton. Already in
1956, Section Nine of the Republican Party platform called for "the
establishment of American air and navy bases all around the world."
This was the party that so many liberal commentators still wrongly
called "isolationist." This was the party to which Patrick Buchanan
adhered and the American foreign policy that he vocally thumped for
until very recently.

The other trouble with Buchanan's anti-imperialist thesis is his
argument that what happened to the British Empire applies obviously
to the present American one. There are two points against this. One
is that history does not repeat itself, and the rise and decline of
Britain's empire was and remains quite different from the American
situation. Buchanan's argument is that the Second World War--more
precisely, Churchill's decision to resist Hitler, no matter what the
cost--was a disaster for Western civilization but, more directly,
for the British Empire itself. Yet the gradual liquidation of the
British Empire, and the piecemeal acceptance by the British people
of that, long preceded World War II.

The further and perhaps deeper problem is Buchanan's sincerity.
Since when has he been an admirer of the British Empire? There is no
evidence for such an affection in his public or writing career until
now. To the contrary, there is ample evidence of his conviction that
the United States should not have supported Britain and its empire
either in the First or in the Second World War.

Here I arrive at the main theme of this book. How Britain Lost Its
Empire and the West Lost the World is only its subtitle, its main
title being Churchill, Hitler, and the Unnecessary War. This
emphasis accords with what is--and has been for a long
time--Buchanan's view of history. The Second World War was an
unnecessary war; a wrong war, especially involving Europe; it was
wrong to fight Hitler; and Churchill was primarily, indeed
principally, responsible. A man has, or more precisely chooses, his
opinions. The choice, ever so often, depends on his inclinations. In
this review it is not my proper business to speculate about
Buchanan's inclinations. I must restrict myself to questioning his
arguments.

The British decision to offer an alliance to Poland in 1939 was a
hasty one, replete with unintended consequences. Partly true. Hitler
did not wish to destroy the British Empire. Partly true. He did want
to destroy Communism and the Soviet Union. Partly true. Churchill
was a warrior; he was obsessed with the danger of German power.
Partly true. Hitler wanted to expel Jews from Europe but not to
exterminate them, at least not while the former policy was still
possible. Again, partly true. Or in other words, true but not true
enough. Here is a difference between Patrick Buchanan and David
Irving. The latter employs falsehoods; Buchanan employs half-truths.
But, as Thomas Aquinas once put it, "a half-truth is more dangerous
than a lie."

The Second World War began in September 1939, with Hitler's armies
invading Poland. Buchanan writes that the British commitment to
Poland was a stupid mistake and that the Poles should not have
fought Hitler. Now here is an example of a special pleader's method:
selective quotation. Buchanan will quote A.J.P. Taylor when this
suits him, as when Taylor wrote, "Only Danzig prevented cooperation
between Germany and Poland." (Taylor was wrong: all evidence shows
that what Hitler wanted was a Poland bereft of any independence from
Germany.) Of course, Buchanan will not cite Taylor's four words
describing Churchill: "The savior of England."

Let me now raise the question: What would have happened if Britain
and France had allowed Hitler to conquer Poland? After that he would
have gone further east and then conquered the Soviet Union, with the
acquiescence of the West. All to the good, Buchanan writes, since
Communism was evil, more dangerous than German National Socialism.
But there is--and there ought to be--no comparison here. Germany was
part and parcel of European culture, civilization, and tradition.
Russia was not. Stalin had a predecessor, Ivan the Terrible. Hitler
had none. German National Socialist brutality was unprecedented.
Russian brutality was not. Nationalism, not Communism, was the main
political force in the 20th century, and so it is even now. When the
Third Reich collapsed in 1945, perhaps as many as 10,000 Germans
killed themselves, and not all of these had been Nazis. When the
Soviet Union and Communist rule in Eastern Europe collapsed in 1989,
I do not know of a single Communist, whether in Russia or elsewhere,
who committed suicide.

There was a consistency in Churchill's view of Europe and of the
world. To him, and for Britain, there were only two alternatives:
either all of Europe dominated by Germany or the eastern half of
Europe dominated by Russia, and half--especially the western
half--of Europe was better than none. Besides, Churchill said that
the Russians could swallow Eastern Europe but not digest it and that
Communism would disappear from Eastern Europe before long. If Hitler
had won the war, German rule would have been much more enduring.

This is not the first of Buchanan's many expressions of his visceral
and intellectual antipathy to Churchill. Irving's main method in
defending Hitler is to blacken all of Hitler's opponents, foremost
among them Churchill. But then he is obsessed with what is and what
is not true of the Holocaust. Buchanan is not. In this book,
Buchanan deprecates Hitler: in 1942 "he was absorbed in self-pity:
and he was condemning his own people." On page 383: Hitler's was "an
evil and odious regime." But there is a fatal contradiction in
Buchanan's theses: Hitler's regime--including, one may think, its
expansion--was evil, but warring against him was unnecessary and
wrong. Either thesis may be argued, but not both.     [dingbat.gif]
___________________________________

John Lukacs is author of a number of books, including George Kennan:
A Study of Character and Blood, Toil, Tears and Sweat: The Dire
Warning: Churchill's First Speech as Prime Minister.

cccc
Stephen Pollard: Social mobility disappeared with grammar schools
http://www.timesonline.co.uk/tol/comment/columnists/guest_contributors/article4200681.ece
June 24, 2008

Comprehensives were supposed to bring equality. They did the opposite


It's a puzzle how Gordon Brown manages to maintain the aura of a
serious intellectual. He clearly reads widely. But so, too, do my
nephews, albeit books with shorter words. The problem lies not with
his ability to read but to draw the correct conclusions.

His speech yesterday on social mobility is a case in point - a weird
mix of platitudes and outright nonsense. Parents should want their
children to do better than they did themselves. Wow. What an
insight. And this "cannot be achieved without people themselves
adopting the work ethic, the learning ethic and aiming high... We
must set a national priority to aggressively and relentlessly
develop the potential of the British people." It's difficult to
imagine a priority aggressively and relentlessly to hold back the
potential of the British people.

The difficulties start when he talks in more than platitudes.
Yesterday's speech was predicated on the notion that, while he had
been fortunate to be "a child of the first great wave of postwar
social mobility", there was then a "lost generation" of "Thatcher's
children" who were denied the chance to progress. Mr Brown is right
to talk about the reversal in social mobility that took place in the
last century. But he is about as far from the truth as it is
possible to imagine in describing its cause. Margaret Thatcher did
not create the problem; she inherited it.

A 1996 study by the Institute for Fiscal Studies confirmed what
strikes most people instinctively: education is the great engine of
social mobility. "There is a clear correlation between high mobility
up the income distribution and a high level of educational
attainment. Non-movers are almost five times as likely to have no
qualifications as big movers; at the other end of the scale, big
movers are more than seven times as likely to have A levels or
better than non-movers are." And with the educational opportunities
laid out in Rab Butler's 1944 Education Act, which enshrined the
tripartite system of grammar, technical and secondary modern
schools, increasingly it was no longer true that where you were born
on the social scale determined where you ended up.

It is made clear in the long-awaited circular that all selective
secondary schools should eventually disappear
Leading article: Second best

As Churchill said to the boys of his alma mater, Harrow School, in
1940: "When this war is won... it must be one of our aims to
establish a state of society when the advantages and privileges
which have hitherto been enjoyed by the few shall be more widely
shared by the many, and by the youth of the nation as a whole."

And this started to happen: the proportion of public-school-educated
undergraduates at Oxford was, for instance, on a steady downward
path after the Second World War. In 1946 65 per cent of male
students were from independent schools. By 1967 only 53 per cent of
male students were from public schools. The pattern was even clearer
with women, the share falling from 57 per cent of arts
undergraduates in 1946 to 39 per cent in 1967. For all the problems
with technical and secondary modern schools, grammar schools did a
fine job of lifting children out of poverty and into opportunity.
Yet today, our comprehensive system has one of the worst rankings in
the developed world.

Education was seen by the advocates of comprehensive schools "as a
serious alternative to nationalisation in promoting a more just and
efficient society" (as Tony Crosland, who would not rest until he
had "destroyed every f***ing grammar school", put it). But this was
Grade A drivel. Class divisions were made worse, not better. Now
those who can afford to do so leave the state system for private
education or move to a middle-class catchment area. The rest are
stuck with what they are served up. As A.H.Halsey, an adviser to
Crosland and one of the leading egalitarian theorists of the 1960s,
put it: "The essential fact of 20th-century educational history is
that egalitarian policies have failed."

The speed of the process was astonishing. In the late 1960s the
state grammar schools and quasi-state direct grant schools easily
outclassed the independent sector in terms of academic output. The
next decade saw both these meritocratic pillars of the state school
system collapse. In 1971 35 per cent of all state schools were
comprehensive; in 1981 the figure was 90 per cent, and almost all
the direct grant schools had joined the private sector. In
destroying the direct grant schools on the altar of equal
opportunity, the 1974-79 Labour Government succeeded only in denying
opportunity to many poor children.

Mr Brown is right to emphasise the imperative of social mobility.
But until he stops speaking in platitudes and starts understanding
what has gone wrong, he will never be able to put anything right.

Stephen Pollard is author of A Class Act: the Myth of Britain's
Classless Society

Have your say

30 kids in a compre class does not work, and for all the good will
in the world shoving the special needs kids in main stream may be
nice for them but draggs all the other children down, the only way
forward is selection 100% of the way.
Specail schools for special kids is the way forward!
MR W jones, Liverpool, England

Interestingly almost every comment is from someone who went to
grammar school and then did well. What about all the others (the
vast majority) who didn't have this chance? Your article has a give
away phrase about the problems the other schools had, it was to fix
this that changes were made.
EBS, London, London

If a child is less able, how would they do better in a comprehensive
than at a technical or secondary modern school? Why treat everyone
the same when there is a huge spectrum of ability and talent? Why
not offer better opportunities to people once they leave school such
apprenticeships?
Nicola Mullett, Lightwater, UK

My grandson went to Grammar School and is now an articulate, well
mannered, and hard working young man.
I have no doubt that had he gone to the local comprehensive where
bullying and thuggish gangs were rife, much of the same sort of
behaviour would have rubbed off on him. Thank God he didn't!
wooram, alicante, spain

"For all the problems with technical and secondary modern schools .
. ." Now, that really begs the question. Three-quarters of children
were "educated" in such schools and most left them without any
qualifications. The main purpose of the Tripartite System was to
reinforce existing class divisions.
Stewart, London,

Killer fact - most of the Cabinet were privately educated - that's
all you need to know about modern socialism.
john miller, london, uk

Going to grammar school has been a great help to me. I am totally
against super classes where pupils of all ages will eventually all
sink into the same quagmire. It is not fair to any child of any
ability.
Nat, Kent, UK

Comprehensives bring everybody down to the same low level.
Socialists will never admit this, even though the facts are staring
them in the face.
Jon Leigh, Southern, France

Comprehensive Education is how Public Schoolboys were able to smash
competition from Grammar Schools. Almost every Secretary of State
for Education since 1944 went to Public School and presided over the
destruction of Grammar Schools.
TomTom, Leeds, England

I am from the slums of Birmingham , passed the 11 plus and became a
successful business man. Probably about 60% of my class mates fared
similarly.
In an otherwise excellent article the Thatcher commentc is not
appropriate.
She was in Power long enough to restore the Grammars destroyed by
Labor
Charles Daniels, Lady Lake, Florida

The best laid plans of mice and men and all that... One of the
(many) problems with leftist thinking is that it never looks more
than one step ahead and therefore falls frequent victim to the law
of unintended consequences. Like a chess player capable of playing
only one move at a time. Losers.
Billy Barnett, HK,

Born in 1942, I grew up in a poor, fatherless, family. My mother was
uneducated but believed in education and high moral standards.
Grammar school led us to higher things, with the next generation
going even further; I was an economic adviser to PMs, kids are
doctor, engineer, etc, and fulfilled.
Faustino, Brisbane, Australia

It's wrong to say that 'Margaret Thatcher did not create the
problem'. As Minister of Education in the seventies she approved
more comprehensive reorganization schemes than anyone else.
Paul, Sydney, Australia

The author quotes a study from 1996. He could have cited the 2006
study by the ESRC showing that education policy plays no part in
social mobility. In any case the middle classes have no interest in
downward mobility whatever education system is in place. This pie
has only so many slices.
philip, cambridge,

My father was the youngest of 16 children, all born into poverty. I
was born 1944, the year of Butler's Education Act. I escaped a
similar life of poverty due to 2 life changing events. I was the
first in my family to go to a grammar school & then university & I
left Oldham forever!
Jack, Leatherhead, England

I went to a grammar and have taught in one. Fab, but they are not as
good for social mobility as they used to be, as primaries in working
class areas often ignore the entry tests for ideological reasons.
But new grammars located in poor areas would lift aspirations and
bring in the middle classes.
Jason, Chelmsford, UK

[cont.] "specialist colleges" where schools can select a certain
number on the basis of musical talent or something and call themself
a specialist school! I wouldn't have met many of my grammar school
friends had we lived in another area - they would have been
privately educated. Mobility, eh?
Amy Allen, London,

Everything you say here is correct, except for the suggestion that
this was a case of error. Just read the quote from Crosland.
The destruction of education was about as much a mistake as Mao's
Great Leap Forward.
In each case the aim was to sacrifice results in pursuit of equality
of poverty.
jon livesey, Sunnyvale, CA/USA

wish I had my time again, my career would have turned out so
differently in a grammar school
Instead I went to the local comp where the education was one size
fits all approach. Bored rigid with the regime left 5 yrs later as
one of the biggest under acheivers of 1979
karen, cheshire, uk

The comprehensive principle is now being applied to the university
sector where, before long, 50% of young people will take degrees.
But there are already too many students who lack the ability to
study at that level. It's not their fault, but the universities
encourage them in order to earn fees.
Dr. Denis MacEoin, Newcastle upon Tyne, UK

About time someone spoke up! My grammar school turned comprehensive
after one year in 1971. The result was a large comprehensive in
which lack of order allowed bullying and general disorder such that
I could not wait to leave. Comprehensive drags down good pupils - it
does not pull up.
Richard, Plymouth,

Absolutely right. The situation was made worse by the creation of
'mass' higher education, and the consequent substitution of the
student loan, with its ever-increasing burden of debt, for the
student grant. Far from increasing social mobility, this policy has
reduced it.
RM Blaber, Wellingborough, UK

I started life in poverty in a rented basement flat with working
class parents and ended my career as a medical professor. I hold the
grammar school responsible.
Terry Hamblin, Bournemouth,

Your article is excellent and accurate, except for one point.
The fault does not stop with Brown, as David Cameron has adopted the
same stance. Even though he knows this is wrong, he, or his
advisors, do not have the balls to do the right thing and risk
losing votes.
Tim Devereux, Esher, Surrey

Well said, but too late I fear. Meritocratic educational
opportunities are the best route off the bottom of the ladder, but a
route denied to recent generations by left wing ideologues. Half
baked social engineering shows the laws of unintended consequences
at their most immutable.
gordon w, Didcot, England

Spot on. And egalitarian policies have failed in the wider sphere
too, by creating a dependency culture. However Gordon need have no
fear, the non-egalitarian nature of the global future will force a
true worldwide meritocracy, and we had better participate!
Paul Freeman, London, England

Couldn't agree more!
40 years ago, my best friend at grammar school was the son of a
shepherd who lived in a tied cottage.
He and I (the son of a managing director) both received the sort of
free education that now costs thousands per term at private school.
Where would my pal go today?
Mike , Newbury, UK

My grammar school education made me see a world my parents had no
knowledge of. I remember my Mum used her redundancy money to get me
the uniform. We are not all academic so why do we force our children
down this route? If Technical education was what it should be, a
grammar school would be what?
Lynda, Bournemouth, Uk

Of course Comprehensives are a rubbish idea. Children can be
academic, or good at technical subjects or science,,or hands on
crafts, or arts, or.... they are fragile and need small classes.
Children should be offered an education to suit them not made to fit
in with someones political dogma.
Angus, Milton Keynes, UK

cccc
Solving postal problem could win million-dollar prize
http://www.newscientist.com/article.ns?id=mg19826611.500&print=true

23 June 2008
Ian Stewart

EVER since a Babylonian scribe decided to teach his students
arithmetic by setting them problems using the formula "I found a
stone but did not weigh it..." mathematicians have celebrated the
hidden depths of apparently everyday problems. They have found
inspiration in slicing pies, tying knots and spinning coins. But
even mathematicians have been surprised by the depth of the mystery
that lurks behind an innocent question about postage stamps.

Suppose that your post office sells stamps with just two values: 2
cents and 5 cents. By combining these values, you can make up almost
any whole number of cents. For example, to post a letter costing 9¢,
you could stick one 5¢ stamp and two 2¢ stamps on the envelope. Two
values that you cannot achieve are 1¢ and 3¢ - and in fact these are
the only impossible amounts. You can produce any even amount using
2¢ stamps - given a big enough envelope - and any odd value from 5¢
upwards, using one 5¢ stamp and multiple 2¢ stamps.

This example is typical. Given an unlimited supply of stamps, there
is always some key value above which any total can be achieved by
sticking the right combination of stamps on the envelope. This is
also true if you have more than two denominations of stamp
available.

But the million-dollar question is this: with n denominations of
stamps available, what is that key value? The first person to
consider a simple version of this question was James Joseph
Sylvester in 1883 (to be precise, he was dealing with coins, but for
our purposes we'll stick with stamps). Sylvester came up with a
simple formula for finding this key value when dealing with just two
denominations (see "Pushing the envelope").

In its general form, the postage-stamp problem really could be a
million-dollar question: the Clay Mathematics Institute in
Cambridge, Massachusetts, is offering exactly that amount to anyone
who can solve a problem that is logically equivalent to it. We now
have tantalising new hints that the postage-stamp problem - and
therefore, perhaps, the related million-dollar enigma - might not be
as daunting as it appears. So considering how to pay for posting our
mail might lead to a breakthrough in one of the most significant
mathematical problems of the 21st century.

It will never compute

The issue centres around the cost of solving a problem - not in
dollars and cents, but in computational effort. We measure the
difficulty of a calculation by the number of basic computational
steps needed to complete it: for a particular size of problem -
often measured in terms of the number of digits in the number to be
crunched - what is the "running time" of the algorithm concerned? If
the problem concerns 50-digit numbers rather than 25-digit numbers,
say, how much longer does the algorithm take to get the answer? What
about 100-digit numbers, or any number of digits? It should be noted
that this running time is an abstract notion, related but not
equivalent to the actual time taken by any given computer.

In broad terms, a computational method is practical - "efficient" or
"easy", if you prefer to look at it that way - if the running time
grows in step with some fixed power of the number of digits required
to pose the question. For example, an algorithm for testing a number
n to see whether it is prime may have a running time linked to the
sixth power of the number of digits of n.

Such algorithms are said to be "class P", where the "P" stands for
"polynomial". Algorithms that run in polynomial time are relatively
stable: they do not get wildly slower with small increases in the
size of the input. In contrast, non-P algorithms are generally
impractical - "inefficient"or "hard" - and become unmanageable with
relatively small increases in input size. It's not quite that
straightforward, because some non-P algorithms are pretty efficient
until the input size gets very big indeed, while some P algorithms
depend on a parameter which isso large that they couldn't actually
run within a human lifetime. Nevertheless, the distinction between P
and non-P seems to be the most basic and important distinction in
problems about the efficiency of algorithms - a way to formalise the
intuitive ideas of "easy to compute" versus "hard to compute".

Are there any such things as truly hard problems? Yes, several
kinds. The obvious ones are hard for a simple reason, such as
printing out the answer takes too long. A good example is "print all
ways to rearrange this list of symbols". With the 52 symbols in a
pack of cards, the list would contain
80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,2
77,824,000,000,000,000 arrangements, and you'd have to print the
lot. These types of problem have to be excluded, which we do by
introducing another class of algorithm, confusingly called NP, which
run in "nondeterministic polynomial" time. A problem is NP if any
proposed solution can be checked to determine whether it is right or
wrong, in polynomial time - that is, in reasonable time. A rough
analogy is solving a jigsaw puzzle. However long it takes to work
out how to fit the pieces together - the nondeterministic aspect - a
brief glance at the result usually reveals whether it is correct.

All this classification has led to a rather fundamental question,
and whoever cracks it will take the Clay prize: is NP really any
different from P? To put it plainly, if it is easy to check the
accuracy of any proposed solution to a problem, must there be an
easy way to solve the problem in the first place?

The smart money says NP problems need not be P: even if it is easy
to check any proposed solution to a problem, you can't solve that
problem efficiently by making repeated guesses and checking them in
turn, because the sheer number of possibilities is too large. Think
of opening a combination lock by trying every combination in turn. A
single satisfying "click" greets the correct answer, but if you are
dealing with a sophisticated lock you could spend a lifetime trying
successive combinations. Guessing at a computer password is another
example.

Even without the Clay prize as motivation, most mathematicians would
sell their mothers into slavery to find out whether NP is distinct
from P because it is such a baffling and fundamental problem. The
truly tantalising thing about this conundrum is that it is an
example of an "NP-complete" problem. NP-complete problems are a
subset of NP problems and are special in that if an efficient
solution to any of them can be found, then that same solution can be
used to solve any NP problem efficiently. In other words, finding an
efficient way to solve any NP-complete problem means we have shown
that all NP problems are effectively P. The matter isn't entirely
esoteric either: the problem is related to some issues in banking
security, for example, so there is good reason to pursue a solution.
Which brings us back to our very own problem - postage stamps.

It was Jorge Ramirez-Alfonsin of the Pierre and Marie Curie
University in Paris, France, who proved in 1996 that the general
version of the stamp problem - with an unlimited number of stamp
denominations - is NP-complete. So if there is an efficient method
to solve the postage-stamp problem, there is an efficient method to
solve any NP problem.

Well, is there an efficient method? Not yet, but there is a new
reason for optimism, at least for a simplified version of the
problem in which there is a limit on the number of stamp
denominations allowed - three, four, 10, 100, whatever - though with
such a limit in place, however large, the problem is no longer
NP-complete. What's more, simple formulae for the postage-stamp
problem of the type that Sylvester found no longer apply. But more
complex algorithms do and in principle they run in polynomial time,
so if we consider such specific examples of our NP-complete problem,
they become class P.

As the number of stamp denominations increases, so will the power of
the input size that determines the running time. Suppose, for
instance, that for three stamps you could find an algorithm whose
running time is proportional to the third power, or cube, of the
total number of digits in the stamps' values; for four stamps you
could find an algorithm whose running time is proportional to the
fourth power of the number of digits and so on. In general, for n
denominations of stamp you could find an algorithm whose running
time is proportional to the nth power of the number of digits. This
is good news: in this scenario, broadly speaking, each separate
3-stamp, 4-stamp or 1000-stamp problem would be class P. Indeed,
Ravi Kannan, now at Yale University, showed in 1992 that there is an
"efficient" solution for each number of stamps.

It is not all good news, however. Ramirez-Alfonsin's theorem shows
that we cannot combine these individual methods to obtain one
algorithm that is efficient no matter how many stamps we have. The
whole lot lumped together into the general n-stamp problem would not
be class P, because there is no upper limit to the powers that could
occur. Furthermore, Kannan's theory turns out to be one of those
cases where the theoretical concept of efficiency does not equate to
"practical": gigantic exponents or multipliers undo much of his good
work. As a result, no one has even considered using his method to
program a computer to solve the problem.

The latest twist in this tale is more encouraging, though. Stan
Wagon of Macalester College in St Paul, Minnesota, and colleagues
can now solve 4-stamp problems that involve 100-digit numbers in
about 1 second on a fast desktop computer, and 10-stamp problems
with 10-digit numbers in two days. Better still, Bjarke Roune at the
University of Aarhus, Denmark, has used computational algebraic
geometry to solve 4-stamp problems with 10,000-digit values in a few
seconds, and 13-stamp problems with 10-digit values in a few days.

While all of this leaves the general n-stamp problem unperturbed, it
proves that an NP-complete problem can be solved in many practical
situations. Imagine if one "nasty" combination of stamps in every
trillion possibilities would take forever to solve, but that all the
others take only a few minutes. The probability of encountering one
of the nasty cases in any particular instance would be negligible.
In this scenario, although the generalised problem is NP-complete
and unavoidably hard, most specific examples of it could be much
easier with the right approach. We already know that something like
this happens for the travelling salesman problem - which aims to
calculate the shortest return journey through n cities, visiting
each city only once - and for some problems in mathematical
economics. Now we're seeing it again for postage stamps. The same
might occur in other NP-complete problems.

There are practical implications to all of this. Even if the method
that your bank uses to encrypt your account information is
NP-complete "in general" - which is more than can currently be
proved for most practical encryption systems - the particular
version that your bank is using might nevertheless be insecure. That
is, perhaps, not a good enough reason to rush off to check your bank
statement, but it could make us rethink the meaning of secure
encryption.

The most exciting possibilities thrown up by the latest approaches
to the postage-stamp problem renew hope for solving many problems
that previously seemed unassailable. By excluding the rare
worst-case scenarios and focusing our attention on the typical ones,
we might lick them yet.

Pushing the envelope

Suppose that you have two types of stamp: 4¢ and 5¢. Which totals
can you achieve, and what is the biggest total you cannot obtain?
Clearly you can get:

4
5
9 = 4+5

10 = 5+5

12 = 4+4+4

13 = 4+4+5

14 = 4+5+5

15 = 5+5+5

16 = 4+4+4+4

17 = 4+4+5+4

and so on. The numbers 1, 2, 3, 6, 7 and 11 are impossible, but as
soon as we can achieve four consecutive whole numbers (12, 13, 14,
15) then we can add extra 4s to them to get 16, 17, 18 and 19, then
20, 21, 22 and 23, and so on - with no gaps. So every value from 12
upwards can be achieved. The largest impossible total is thus 11
(see Diagram).

Now, 11 is equal to (4 × 5) - 4 - 5. This pattern is universal.
Given stamps of denominations x and y, with no common factor (other
than of course 1), the largest total that cannot be achieved is
exactly xy - x - y. So, for instance, if the two values are 99¢ and
101¢, then you can get every total larger than, but not equal to,
(99 × 101) - 99 - 101, which is 9799¢.

Related Articles

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http://www.newscientist.com/article.ns?id=mg17423454.400
01 June 2002
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26 August 2006
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http://www.newscientist.com/article.ns?id=mg18124315.400
24 January 2004
The prime number hunters close in
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Weblinks

The Clay Mathematics Institute
http://www.claymath.org
The P-versus-NP page
http://www.win.tue.nl/~gwoegi/P-versus-NP.htm
Stan Wagon's home page
http://www.stanwagon.com
Ian Stewart's home page
http://www2.warwick.ac.uk/fac/sci/maths/people/staff/Ian_Stewart