
Post by Lonely on Mar 7, 2022 8:04:50 GMT



Post by Lonely on Mar 7, 2022 22:41:10 GMT
Solution # 1/ 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.
Solution # 2/ The 24 rolls are in a circle. Draw an arbitrary line that cuts the circle in half, 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 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 decreases 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.

