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Wired have come up with an interesting conundrum which will keep some of you (and me) thinking for a while:
In 1850, the Reverend Thomas Kirkman, rector of the parish of Croft-with-Southworth in Lancashire, England, posed an innocent-looking puzzle in the Lady’s and Gentleman’s Diary, a recreational mathematics journal:
“Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily, so that no two shall walk twice abreast.” (By “abreast,” Kirkman meant “in a group,” so the girls are walking out in groups of three, and each pair of girls should be in the same group just once.)
Pull out a pencil and paper, and you’ll quickly find that the problem is harder than it looks: After arranging the schoolgirls for the first two or three days, you’ll almost inevitably have painted yourself into a corner, and have to undo your work.
[...]
Yet for more than 150 years after Kirkman circulated his schoolgirl problem, the most fundamental question in the field remained unanswered: Do such puzzles usually have solutions? Kirkman’s puzzle is a prototype for a more general problem: If you have n schoolgirls, can you create groups of size k such that each smaller set of size t appears in just one of the larger groups? Such an arrangement is called an (n, k, t) design. (Kirkman’s setup has the additional wrinkle that the groups must be sortable into “days.”)
It’s easy to see that not all choices of n, k and t will work. If you have six schoolgirls, for instance, you can’t make a collection of schoolgirl triples in which every possible pair appears exactly once: Each triple that included “Annabel” would contain two pairs involving her, but Annabel belongs to five pairs, and five is not divisible by two. Many combinations of n, k and t are instantly ruled out by these sorts of divisibility obstacles.
For the parameters that aren’t ruled out, there’s no royal road to finding designs. In many cases, mathematicians have found designs, through a combination of brute force and algebraic methods. But design theorists have also found examples of parameters, such as (43, 7, 2), that have no designs even though all the divisibility requirements check out. Are such cases the exception, mathematicians wondered, or the rule? “It was one of the most famous problems in combinatorics,” said Gil Kalai, a mathematician at the Hebrew University of Jerusalem. He recalls debating the question with a colleague a year and a half ago, and concluding that “we’ll never know the answer, because it’s clearly too hard.”
But he was wrong .... Read the full linked article for more.
Original Submission
(Score: 0) by Anonymous Coward on Monday June 22 2015, @01:17PM
Abreast doesn't mean "in a group", it means "Side-by-side". Imagine school girls walking in a big group, in lines of 3 wide. Common language in the UK, presumably not in the US?
(Score: 0) by Anonymous Coward on Monday June 22 2015, @01:37PM
I got it. But I am also letting my inner 13 year old out today :)
(Score: 2) by hemocyanin on Monday June 22 2015, @01:57PM
Totally common word in the US. The incorrect definition in TFA was totally unnecessary.
(Score: 4, Insightful) by VLM on Monday June 22 2015, @02:15PM
Its a trick question.
posed an innocent-looking puzzle
Yeah whatever. They're using words like:
young ladies
and
abreast
to distract us. Just rephrase the question to something like scheduling first person shooter videogame multi player maps such that each 3 map set doesn't have repeats... then you'll get a much less distracted answer.
(Score: 2) by Freeman on Tuesday June 23 2015, @07:29PM
If someone had made a computer game out of it, it could have been solved 20 years ago.
Joshua 1:9 "Be strong and of a good courage; be not afraid, neither be thou dismayed: for the Lord thy God is with thee"
(Score: 2) by wonkey_monkey on Monday June 22 2015, @03:12PM
It's a verbatim quote of the original puzzle, with the clarification to explain what the creator actually meant.
systemd is Roko's Basilisk
(Score: 0) by Anonymous Coward on Monday June 22 2015, @02:29PM
It is not common in the US to say 'abreast'. You are correct that 'abreast' means 'side-by-side' but in the context of the puzzle, it is correct to paraphrase 'in groups of three'.
(Score: 0) by Anonymous Coward on Monday June 22 2015, @02:45PM
> It is not common in the US to say 'abreast'.
I guess it all depends on what circles you move in.
Your not common is my unremarkable.
(Score: 0) by Anonymous Coward on Tuesday June 23 2015, @08:39AM
Sorry, I was expelled from the Pedantry Squad at Mensa.
Now I get to talk with my neighbors: 15% teenage mothers, 46% former teenage mothers, 9% teenage fathers, 30% ex-convicts.