Can you solve it? The maths of Lviv

I have not been able to think about anything else other than the war in Ukraine. The history of 20th century mathematics was influenced by the western city of Lviv. A group of scholars came up with new ideas and methods that helped shape the subject for decades.

Stanis Ulam and Hugo Steinhaus were two of the Polish mathematicians who studied at the LwF3;w school of mathematics.

These friends were unconventional at work. A thick notebook was set aside to record important problems and solutions, as much of their time was spent in cafés. The Scottish book has a legendary status in the mathematical community, having survived the Second World War and inspiring further research. Some of its problems are still open.

One of the best known ideas introduced by the LwF3;w school is the ham sandwich theorem, which was solved by Banach using a result of Ulam. 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.

Today's puzzles are about dividing food. The first is from Hugo Steinhaus. The second uses a method that is related to the ham sandwich theorem.

Three friends contribute to buy a ham. The first friend says the weights are equal. The distrustful of the first friend reweighs the pieces and judges them to be worth something. The distrustful of them both, weighed the ham on their own scales, getting another result.

If each friend insists that their weighings are correct, how can they share the pieces without having to admit they got at least four hams?

The rolls are arranged in a circle. It is possible to divide the circle into two halves by using a straight line, as there are an equal number of plain and seeded rolls on either side of the line.

I will be back at 5pm UK with the solutions.

Please don't give away any information.

The Borsuk-Ulam Theorem and the bisection of necklaces are references in the question 2 that was adapted from Mathematical Puzzles.

On Mondays, I set a puzzle here. I always look for great puzzles. Email me if you would like to suggest one.

I'm the author of several books of puzzles, most recently the Language Lovers Puzzle Book. I give school talks about math and puzzles. Please contact your school if you are interested.

I will be giving a puzzles workshop on April 21. You can sign up here.