Like lots of you I’ve hardly been in a position to consider anything these previous ten days aside from the battle in Ukraine. So at present’s puzzles are a celebration of Lviv, Ukraine’s western metropolis, which performed an necessary function within the historical past of twentieth century arithmetic. During the Thirties, a outstanding group of students got here up with new concepts, strategies and theorems that helped form the topic for many years.
The Lwów faculty of arithmetic – at the moment, the town was in Poland – was a closely-knit circle of Polish mathematicians, together with Stefan Banach, Stanisław Ulam and Hugo Steinhaus, who made necessary contributions to areas together with set-theory, topology and evaluation.
These pals had unconventional working habits. Much of their time was spent in cafés, particularly the Scottish Café, the place a thick pocket book was put aside to report necessary issues and options. The so-called “Scottish book” has a legendary standing within the mathematical neighborhood, surviving the Second World War, and finally inspiring additional analysis. Some of its 200-odd issues stay open.
Of the numerous concepts launched by the Lwów faculty, top-of-the-line identified is the “ham sandwich theorem,” posed by Steinhaus and solved by Banach utilizing a results of Ulam’s. It states that it’s potential to slice a ham sandwich in two with a single slice that cuts every slice of bread and the ham into two equal sizes, regardless of the measurement and positions of the bread and the ham.
Today’s puzzles are additionally about dividing meals. The first is from Hugo Steinhaus’ One Hundred Problems in Elementary Mathematics, printed in 1938. The second makes use of a technique concerned within the proof of the ham sandwich theorem.
1) Three pals every contribute £4 to purchase a £12 ham. The first good friend divides it into three components, asserting the weights are equal. The second good friend, distrustful of the primary, reweighs the items and judges them to be value £3, £4 and £5. The third, distrustful of them each, weighs the ham on their very own scales, getting one other outcome.
If every good friend insists that their weighings are right, how can they share the items (with out chopping them anew) in such a approach that every of them must admit they acquired at the least £4 of ham?
2) Ten plain and 14 seeded rolls are randomly organized in a circle, equidistantly spaced, as beneath. Show that utilizing a straight line it’s potential to divide the circle into two halves such that there are an equal variety of plain and seeded rolls on both facet of the road.
I’ll be again at 5pm UK with the options.
PLEASE NO SPOILERS
Question 2 is tailored from Mathematical Puzzles by Peter Winkler, who offers as a reference Alon and West, The Borsuk-Ulam Theorem and bisection of necklaces, Proceedings of the American Mathematical Society 98 (1986).
I set a puzzle right here each two weeks on a Monday. I’m all the time on the look-out for excellent puzzles. If you wish to recommend one, e-mail me.
I’m the creator of a number of books of puzzles, most not too long ago the Language Lover’s Puzzle Book. I additionally give faculty talks about maths and puzzles (on-line and in individual). If your faculty is please get in contact.
On Thursday 21 April I’ll be giving a puzzles workshop for Guardian Masterclasses. You can join right here.