{"id":2811,"date":"2022-07-30T08:07:30","date_gmt":"2022-07-30T12:07:30","guid":{"rendered":"https:\/\/josmfs.net\/?p=2811"},"modified":"2022-07-30T08:07:30","modified_gmt":"2022-07-30T12:07:30","slug":"the-maths-of-lviv","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2022\/07\/30\/the-maths-of-lviv\/","title":{"rendered":"The Maths of Lviv"},"content":{"rendered":"

\"\"Unfortunately Ukraine has receded from our attention under the threat from our own anti-democratic forces, but this Monday Puzzle<\/a> from Alex Bellos in March is a timely reminder of the mathematical significance of that country.<\/p>\n

\u201cLike many of you I\u2019ve hardly been able to think about anything else these past ten days apart from the war in Ukraine. So today\u2019s puzzles are a celebration of Lviv, Ukraine\u2019s western city, which played an important role in the history of 20th century mathematics. During the 1930s, a remarkable group of scholars came up with new ideas, methods and theorems that helped shape the subject for decades.<\/p>\n

The Lw\u00f3w school of mathematics \u2013 at that time, the city was in Poland \u2013 was a closely-knit circle of Polish mathematicians, including Stefan Banach, Stanis\u0142aw Ulam and Hugo Steinhaus, who made important contributions to areas including set-theory, topology and analysis. \u2026<\/p>\n

Of the many ideas introduced by the Lw\u00f3w school, one of the best known is the \u201cham sandwich theorem,\u201d posed by Steinhaus and solved by Banach using a result of Ulam\u2019s. It states that it is possible to slice a ham sandwich in two with a single slice that cuts each slice of bread and the ham into two equal sizes, whatever the size and positions of the bread and the ham.<\/p>\n

Today\u2019s puzzles are also about dividing food. The first is from Hugo Steinhaus\u2019 One Hundred Problems in Elementary Mathematics<\/em>, published in 1938. The second uses a method involved in the proof of the ham sandwich theorem.<\/p>\n

    \n
  1. Three friends each contribute \u00a34 to buy a \u00a312 ham. The first friend divides it into three parts, asserting the weights are equal. The second friend, distrustful of the first, reweighs the pieces and judges them to be worth \u00a33, \u00a34 and \u00a35. The third, distrustful of them both, weighs the ham on their own scales, getting another result. <\/em>If each friend insists that their weighings are correct, how can they share the pieces (without cutting them anew) in such a way that each of them would have to admit they got at least \u00a34 of ham?_<\/span>\"\"<\/em><\/li>\n
  2. Ten plain and 14 seeded rolls are randomly arranged in a circle, equidistantly spaced, as below. Show that using a straight line it is possible to divide the circle into two halves such that there are an equal number of plain and seeded rolls on either side of the line.<\/em><\/li>\n<\/ol>\n

    Show there is always a diameter that cuts the circle into two batches of 12 rolls with an equal number of plain and seeded.<\/p>\n

    Question 2 is adapted from Mathematical Puzzles by Peter Winkler, who gives as a reference Alon and West, The Borsuk-Ulam Theorem and bisection of necklaces, Proceedings of the American Mathematical Society 98 (1986).<\/em>\u201d<\/p>\n

    See The Maths of Lviv<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

    Unfortunately Ukraine has receded from our attention under the threat from our own anti-democratic forces, but this Monday Puzzle from Alex Bellos in March is a timely reminder of the mathematical significance of that country. \u201cLike many of you I\u2019ve hardly been able to think about anything else these past ten days apart from the […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[15,174],"class_list":["post-2811","post","type-post","status-publish","format-standard","hentry","category-puzzles-and-problems","tag-alex-bellos","tag-logic"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/2811","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/comments?post=2811"}],"version-history":[{"count":5,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/2811\/revisions"}],"predecessor-version":[{"id":2816,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/2811\/revisions\/2816"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=2811"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=2811"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=2811"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}