Related papers: Guess my number! From binary tricks to general bas…
We develop and deploy a set of constraints for the purpose of calculating minimal sizes of lottery designs. Specifically, we find the minimum number of tickets of size six which are needed to match at least two balls on any draw of size…
Let b $\ge$ 2 be an integer and let s b (n) denote the sum of the digits of the representation of an integer n in base b. For sufficiently large N , one has Card{n $\le$ N : |s 3 (n) -- s 2 (n)| $\le$ 0.1457205 log n} \textgreater{} N…
We consider how to make probability forecasts of binary labels. Our main mathematical result is that for any continuous gambling strategy used for detecting disagreement between the forecasts and the actual labels, there exists a…
We introduce a new virtual environment for simulating a card game known as "Big 2". This is a four-player game of imperfect information with a relatively complicated action space (being allowed to play 1,2,3,4 or 5 card combinations from an…
For a given number of colors, $s$, the guessing number of a graph is the (base $s$) logarithm of the cardinality of the largest family of colorings of the vertex set of the graph such that the color of each vertex can be determined from the…
Consider the following hat guessing game: $n$ players are placed on $n$ vertices of a graph, each wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but…
The multiplication game is a two-person game in which each player chooses a positive integer without knowledge of the other player's number. The two numbers are then multiplied together and the first digit of the product determines the…
The deck of a graph $G$ is given by the multiset of (unlabelled) subgraphs $\{G-v:v\in V(G)\}$. The subgraphs $G-v$ are referred to as the cards of $G$. Brown and Fenner recently showed that, for $n\geq29$, the number of edges of a graph…
A defining feature of collectable card games is the deck building process prior to actual gameplay, in which players form their decks according to some restrictions. Learning to build decks is difficult for players and models alike due to…
Consider an n by n array of cards shuffled in the following manner. An element x of the array is chosen uniformly at random; Then with probability 1/2 the rectangle of cards above and to the left of x is rotated 180 degrees, and with…
The 21 card trick is well known. It was recently shown in an episode of the popular YouTube channel Numberphile. In that trick, the audience is asked to remember a card, and through a series of steps, the magician is able to find the card.…
Mirror games were invented by Garg and Schnieder (ITCS 2019). Alice and Bob take turns (with Alice playing first) in declaring numbers from the set {1,2, ...2n}. If a player picks a number that was previously played, that player loses and…
By a well-known result of Bayer and Diaconis, the maximum entropy model of the common riffle shuffle implies that the number of riffle shuffles necessary to mix a standard deck of 52 cards is either 7 or 11--with the former number applying…
Several variations of hat guessing games have been popularly discussed in recreational mathematics. In a typical hat guessing game, after initially coordinating a strategy, each of $n$ players is assigned a hat from a given color set.…
The game "Spot It!" is played with a deck of cards in which every pair of cards has exactly one matching symbol and the aim is to be the fastest at finding the match. It is known that finite projective planes correspond to decks in which…
This paper is about the following question: How many riffle shuffles mix a deck of card for games such as blackjack and bridge? An object that comes up in answering this question is the descent polynomial associated with pairs of decks,…
The purpose of this paper is to introduce the idea of triangular Ramsey numbers and provide values as well as upper and lower bounds for them. To do this, the combinatorial game Mines is introduced; after some necessary theorems about…
In an introductory paper on dollar game played on a graph, Matt Baker wrote the following: ``The total number of borrowing moves required to win the game when playing the 'borrowing binge strategy' is independent of which borrowing moves…
We close the gap in the proof (published by Chen and Lin) of formulas for the minimum number of questions required in the expected case for Mastermind and its variant called AB game, where both games are played with two pegs and $n$ colors.…
In 1998, Ciucu published "No-feedback card guessing for dovetail shuffles", an article which gives the optimal guessing strategy for $n$ cards ($n$ even) after $k$ riffle shuffles whenever $k>2\log_{2}\left(n\right)$. We discuss in this…