Related papers: The odd Hummer principle
A deck of $n$ cards are shuffled by repeatedly taking off the top card, flipping it with probability $1/2$, and inserting it back into the deck at a random position. This process can be considered as a Markov chain on the group $B_n$ of…
It is well known that in games with imperfect information, such as poker, bluffing with some probability can be a component of the optimal strategy. However, as far as we know, nobody has ever exhibited a Scrabble position in which the…
We study a combinatorial game derived from a problem in the German National Mathematics Competition. In this game, two players take turns removing numbers from a finite set of natural numbers, aiming to satisfy a certain divisibility…
The Game of Poker Chips, Dominoes and Survival fosters team building and high level cooperation in large groups, and is a tool applied in management training exercises. Each player, initially given two colored poker chips, is allowed to…
The classical Duhamel principle, established nearly 200 years ago by Jean-Marie-Constant Duhamel, reduces the Cauchy problem for an inhomogeneous partial differential equation to the Cauchy problem for the corresponding homogeneous…
In his book "Mathematical Mind-Benders", Peter Winkler poses the following open problem, originally due to the first author: "[In the game Peer Pressure,] two players are dealt some number of cards, initially face up, each card carrying a…
It is pointed out that an exactly solvable permutation operator, viewed as the quantization of cyclic shifts, is useful in constructing a basis in which to study the quantum baker's map, a paradigm system of quantum chaos. In the basis of…
We consider Flipping Coins, a partizan version of the impartial game Turning Turtles, played on lines of coins. We show the values of this game are numbers, and these are found by first applying a reduction, then decomposing the position…
We investigate the $k$-cycle shuffle on repeated cards, namely on a deck consisting of $l$ identical copies of each of $m$ card types, with total size $n=ml$. We establish asymptotic results for the total variation mixing of this shuffle,…
Inspired by the theory of desirable gambles that is used to model uncertainty in the field of imprecise probabilities, I present a theory of desirable things. Its aim is to model a subject's beliefs about which things are desirable. What…
We give a beautiful explicit example of a convex plane curve such that the outer billiard has a given finite number of invariant curves. Moreover, the dynamics on these curves is a standard shift. This example can be considered as an outer…
We introduce a new type of card shuffle called one-sided transpositions. At each step a card is chosen uniformly from the pack and then transposed with another card chosen uniformly from below it. This defines a random walk on the symmetric…
Poker is a multiplayer game of imperfect information and has been widely studied in game theory. Many popular variants of poker (e.g., Texas Hold'em and Omaha) at the edge of modern game theory research are large games. However, even toy…
Following the work of Lloyd Shapley on the Shapley value, and tangentially the work of Guillermo Owen, we offer an alternative non-probabilistic formulation of part of the work of Robert J. Weber in his 1978 paper "Probabilistic values for…
In each step of the overlapping cycles shuffle on $n$ cards, a fair coin is flipped which determines whether the $m$th card or the $n$th card is moved to the top of the deck. Angel, Peres, and Wilson showed the following interesting fact:…
We study a random game in which two players in turn play a fixed number of moves. For each move, there are two possible choices. To each possible outcome of the game we assign a winner in an i.i.d. fashion with a fixed parameter p. In the…
Apportionment is the problem of distributing $h$ indivisible seats across states in proportion to the states' populations. In the context of the US House of Representatives, this problem has a rich history and is a prime example of…
It is proved that in card games similar to 'Beggar-my-neighbour' the mathematical expectation of the playing time is finite, provided that the player who starts the round is determined randomly and the deck is shuffled when the trick is…
By using a jump transformation associated to the Romik map, we define a new continued fraction algorithm called odd-odd continued fraction, whose principal convergents are rational numbers of odd denominators and odd numerators. Among…
This work arose from efforts to generalise the usual cubical boundary by using different 'weights' for opposite faces, but still to obtain a chain complex, and this method was found to generalise. We describe a variant of the classical…