21 September 2019 | New Scientist | 13Infinity
Chelsea Whyte
IN AN infinite lottery, can you
create a lottery ticket that always
wins? This is the idea behind
a 50-year-old maths problem
that has now been solved.
In a standard lottery, you have
a ticket with a handful of numbers
on it and if they match the randomly
selected numbers from the lottery,
your ticket wins.
Each ticket can have several
rows on it, giving you several
chances to win. This means that
a long enough ticket could in
principle have every possible
winning combination, so always
wins. It would cost so much money
to do this in reality, however,
that it wouldn’t be worth it.
Can the world
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From renewable energy and
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to make energy cleaner and better.Natural gas burns 50% cleaner than coal in power generation.We see possibilities
everywhere.Mathematicians crack
50-year-old problem
The normal
lottery is
somewhat
easier to
win than an
infinite lotteryGETTY IMAGESBut in a hypothetical infinite
lottery, things are a little different.
The winning collection of numbers
is infinitely long, and each ticket can
have an infinite number of rows,
with each row containing an infinite
number of numbers. In fact, the
ticket can be so large that the rows
can’t even be numbered, which
is called being uncountable.
In this situation, it is far less
obvious whether it is possible to
create a ticket that always wins.
It is now half a century since
mathematician Adrian R. D. Mathias
first posed the question, and David
Schrittesser and Asger Törnquist
at the University of Copenhagen
in Denmark have found an answer:it isn’t possible to have a ticket that
always wins the infinite lottery.
About 20 years ago, some
mathematicians rediscovered
the problem and started to make
progress. “Nobody took the
slightest notice for 30 years,
and then suddenly people got
interested again. It’s very satisfying
to see,” says Mathias.It has taken Schrittesser
and Törnquist four years to
solve the puzzle.
The pair used ideas from Ramsey
theory to tackle the problem, a part
of mathematics that looks at how
order appears in a large structure.
They found that in an infinite
lottery, a sort of structure arises that
means the winning numbers clump
together, but in a way that means
a ticket that always wins just can’t
exist (PNAS, doi.org/dbjk).
“With these kinds of problems,
you don’t sit down and say I’m
going to be the one who solves
it, because everyone has tried,”
says Schrittesser. “There’s a little
bit of serendipity.” ❚Can we use maths to clean our oceans?
Find out from Tom Crawford on 11 October
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