https://padlet.com/tpbrzustowski/lhoz061d9egjk3an en-us 2020-12-02 17:18:03 UTC 2025-11-19 16:23:53 UTC hello@padlet.com tpbrzustowski https://padlet.com/tpbrzustowski/lhoz061d9egjk3an/wish/981177409 Consider a n by n checkerboard. At date 0, you can choose to place presents on any squares you want, so that each square contains either 0 or 1 present. At date 1, Santa will add a present in all empty squares that have at least two non-empty neighbours, and repeat this process at dates 2, 3, and so on. What is the minimum number of presents you need to place initially to guarantee that all squares will eventually contain a present? (Note: two squares are neighbours if they have a common side. Thus a given square has two, three or four neighbours, depending on its location on the board).]]> 2020-12-02 18:41:15 UTC https://padlet.com/tpbrzustowski/lhoz061d9egjk3an/wish/981177409 tpbrzustowski https://padlet.com/tpbrzustowski/lhoz061d9egjk3an/wish/981274633 You need to carry out a fair random draw between two options, but your only coin is biased. What can you do? ]]> 2020-12-02 19:01:22 UTC https://padlet.com/tpbrzustowski/lhoz061d9egjk3an/wish/981274633 tpbrzustowski https://padlet.com/tpbrzustowski/lhoz061d9egjk3an/wish/981345018 There are countably many wrapped christmas presents in front of you. You can select one. If you guess correctly what the present is, you can keep it. Before making your guess, you are allowed to open as many of the other presents as you want. Can you come up with a guess that guarantees that you will almost surely keep the present?]]> 2020-12-02 19:16:39 UTC https://padlet.com/tpbrzustowski/lhoz061d9egjk3an/wish/981345018 https://padlet.com/tpbrzustowski/lhoz061d9egjk3an/wish/1008420848 There are N hungry lions and an antelope. The first lion is well positioned to attack and kill the antelope, tiring itself in the process. A tired first lion is the second lion's prey, who may attack and kill the first lion in order to recover from it the dead antelope. Similarly, the second lion tires over fighting, and becomes the third lion's prey. The third lion may thus attack and kill the second lion in order to recover from it the dead antelope, tiring itself in the process. And so it continues. None of the lions is suicidal and prefers to remain hungry than be dead. If not killed by the successive lion, the tired lion gets to eat the antelope instead, and couldn't be more satisfied. Question: when does the first lion kill the antelope?]]> 2020-12-11 00:15:52 UTC https://padlet.com/tpbrzustowski/lhoz061d9egjk3an/wish/1008420848 https://padlet.com/tpbrzustowski/lhoz061d9egjk3an/wish/1008553637 There are two doctors and three shifts, called Monday, Tuesday, and Wednesday, in short M,T,W. Each shift requires the presence of exactly one doctor and, out of fairness, each doctor works at least one shift. Finally, working an additional shift contributes to a doctor's unhappiness (so that M is preferred over both MT and MW but not necessarily over TW). Both doctors know each others' preferences over shifts (no indifferences).

1. Does there exist a Nash implementable SCR which is non-dictatorial according to which doctor 1 works M,T, or W and doctor 2 works TW,MW,MT? Hint: apply a famous theorem.

2. Propose a non-dictatorial SCR and the (sequential) mechanism that implements it. Hint: divide and choose.]]> 2020-12-11 01:21:09 UTC https://padlet.com/tpbrzustowski/lhoz061d9egjk3an/wish/1008553637 https://padlet.com/tpbrzustowski/lhoz061d9egjk3an/wish/1009347914 There are N individuals that seek to pairwise match with one another. Individuals are ranked from 1 to N. Sadly, none of the individuals knows their own rank, but observes everyone else’s rank. Endowed with an infinite amount of time and only concerned with match outcomes—matching with k+1 is unanimously preferred over matching with k over matching with none—propose a matching algorithm whereby pairs (N,N-1),(N-1,N-2),… match with one another.]]> 2020-12-11 10:51:54 UTC https://padlet.com/tpbrzustowski/lhoz061d9egjk3an/wish/1009347914 https://padlet.com/tpbrzustowski/lhoz061d9egjk3an/wish/1009460779
I am late arriving into the airport and have: first to walk to the Shuttle: then wait for the Shittle to depart for the North Terminal; finally walk to the departure area.

Because I am unfit, I can only run (as opposed
to walk) for a single short spurt of time.  Should I run before or after the Shuttle?]]> 2020-12-11 12:11:35 UTC https://padlet.com/tpbrzustowski/lhoz061d9egjk3an/wish/1009460779 https://padlet.com/tpbrzustowski/lhoz061d9egjk3an/wish/1009475216 You have a (large) piece of paper with two crosses on it. Using only a pair of compasses (you have no ruler), find their exact midpoint by drawing seven circles.]]> 2020-12-11 12:22:09 UTC https://padlet.com/tpbrzustowski/lhoz061d9egjk3an/wish/1009475216 https://padlet.com/tpbrzustowski/lhoz061d9egjk3an/wish/1009486148 A rectangle is a Sentez rectangle if two (or four) of its sides are whole multiples of a metre in length.

A rectangle R is covered by a finite number of (non-overlapping) tiles, each tile being a Sentez rectangle.  Show that R, too, is a Sentez rectangle.]]> 2020-12-11 12:29:29 UTC https://padlet.com/tpbrzustowski/lhoz061d9egjk3an/wish/1009486148 https://padlet.com/tpbrzustowski/lhoz061d9egjk3an/wish/1009501743 Count the number of partitions of {1,...,N} such that every partition element is of the form {m,...,n}, i.e., has no wholes.]]> 2020-12-11 12:39:13 UTC https://padlet.com/tpbrzustowski/lhoz061d9egjk3an/wish/1009501743 https://padlet.com/tpbrzustowski/lhoz061d9egjk3an/wish/1009915566 2020-12-11 14:51:25 UTC https://padlet.com/tpbrzustowski/lhoz061d9egjk3an/wish/1009915566