Earlier today I set you the following puzzles about dividing up food, inspired by the Lwów school of mathematics, a group of mathematicians who lived in the city that is now Lviv in the 1930s.
1) Three friends each contribute £4 to buy a £12 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 £3, £4 and £5. The third, distrustful of them both, weighs 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 cutting them anew) in such a way that each of them would have to admit they got at least £4 of ham?
Solution
The third friend must believe that at least one of the pieces is worth at least £4, since if they didn’t, the total of all the pieces wouldn’t add up to £12. They take this piece. There are now two pieces left. One of these pieces must be a piece that the second friend thinks is worth £4 or £5, and so the second friend takes this one. The first friend takes the remaining piece, which they think is worth £4.
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 always 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.
Solution
The 24 rolls are in a circle. Draw an arbitrary line that cuts the circle in half, as below, dividing the rolls into two sides of 12. If one side has five plain rolls, then we’re done. Let’s say the left side has more than five plain rolls. (The right side, therefore, has less than five plain rolls).
Rotate the cut anticlockwise by 1/24 of a full rotation, as shown by the dotted line below, so that the left half loses its roll at the top and gains a roll at the bottom. If the lost and gained roll are the same type, then this half has the same number of plain rolls in total as it did before. If the top one is plain and the other seeded (or vice versa) then the total number of plain rolls reduces by 1 (or increases by 1).
If we keep on rotating the line, each time losing and gaining a roll, then after 180 degrees, the section that was ‘left half’ will now be the ‘right half’, which we know has less than five plain rolls. In other words, the section started with more than 5 plain rolls, and then in a process of either adding or losing a single roll, ended up with less than 5 plain rolls. There must have been a step on the way that there were exactly five plain rolls.
For readers wanting to know what this has got to the ham sandwich theorem (which I discussed in the earlier post), both of them involve applications of the intermediate value theorem, which states that if a value is moving continuously from A to B, it will pass every value from A to B.
I hope you enjoyed today’s puzzle, I’ll be back in two weeks.
I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
I’m the author of several books of puzzles, most recently the Language Lover’s Puzzle Book. I also give school talks about maths and puzzles (online and in person). If your school is interested please get in touch.
On Thursday 21 April I’ll be giving a puzzles workshop for Guardian Masterclasses. You can sign up here.
