Boris Konev and Alexei Lisitsa of the University of Liverpool have been looking at sequences of $+1$s and $-1$s, and have shown using an SAT-solver-based proof that every sequence of $1161$ or more elements has a subsequence which sums to at least $2$. This extends the existing long-known result that every such sequence of $12$ or more elements has a subsequence which sums to at least $1$, and constitutes a proof of Erdős’s discrepancy problem for $C \leq 2$.
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The Chris Tarrant Problem
This is a puzzle I presented at the MathsJam conference. It’s a problem that gave me a headache for a week or so, and I thought others might enjoy it, too. I do know the answer, but I’m not going to give it away — you can tweet me @icecolbeveridge if you want to discuss your theories! (As Colin Wright says: don’t tell people the answer).
You’ve heard of the Monty Hall Problem, right?
Not mentioned recently on The Aperiodical
Summer is a busy time for this site’s hard-working triumvirate, so we haven’t been keeping on top of the news as much as we’d like. There’s been some quite interesting news, so here’s a quick round-up of the most important bits:
Cushing your luck: properties of randomly chosen numbers
Long-time Aperiodical muse David Cushing has made a bet with us that he can give us an interesting post every Friday for the next ten weeks. Every week that he sends a post, we buy him a bar of chocolate. Every week that he doesn’t send us a post, he buys us a bar of chocolate. For his first trick, David is going to do some unnatural things with the natural numbers.
The greatest common divisor (gcd) of two or more integers is the greatest integer that evenly divides those integers. For example, the gcd of $8$ and $12$ is $4$ (usually written as $\gcd(8,12)=4$). Two integers are called coprime (or “relatively prime”) if their gcd is equal to $1$.
A reasonable question to ask is,
Given two randomly chosen integers $a$ and $b$, what is the probability that $\gcd(a,b)=1$?
Erdős Centennial conference
The Hungarian Academy of Sciences, the Alfréd Rényi Institute of Mathematics, the Eötvös Loránd University and the János Bolyai Mathematical Society have announced a conference dedicated to the 100th anniversary of Paul Erdős from 1st-5th July 2013 in Budapest, Hungary.
Porl Air-dursh: a public service announcement
Since we’re all talking and writing about Paul Erdős today, I just thought I’d make a little post clearing up how to write, and how to say, his name.
f(Erdős) = 100
Today is the 100th anniversary of the birth of Paul Erdős, or as most people would call it, Erdős’ 100th birthday. So, Happy Birthday Paul. And if you’ve never heard of him, let’s see what people at his birthday party are saying about the Man Who Loved Only Numbers. Please note: all birthday parties are strictly fictional.
Probably the greatest mathematician of the twentieth century, Paul Erdős … was so eccentric that he made Einstein look normal. He was 11 before he ever tied his shoes, 21 before he ever buttered toast, and died without ever boiling an egg. Erdős lived on the road, traveling from conference to conference, owning nothing but math notebooks and a suitcase or two. His life consisted of math, nothing else.
– Clifford Goldstein, in The Mules That Angels Ride (2005), p. 125