1. A game consists of equilateral triangular tiles with a number from the set {0, 1,2, 3, 4,5} in each of the three corners of each tile (see the figure below). A tile can includerepeated numbers and there are no copies of the same tile in the game (two tiles T and T2 are considered identical when T can be rotated to produce T2).1(a) What is the maximum number of unique tiles that this game can contain?The game includes all possible tiles except for those where all three numbers form a strictly decreasing sequence when read clockwise from the largest number present. Forexample, the tile below is not included.(b) How many tiles are included in this board game?32. A player can place one of their tiles alongside another tile on the table if both pairs of numbers on the sides of the tiles that are touching match. See the figure below for anexample.133.(c) If the tile consisting of three 5's is played at the beginning of the game, how many tiles can be validly placed next to it in the next turn?(d) If the tile consisting of 1, 2, 3, read clockwise around the tile, is played at the beginning of the game, how many tiles can be validly placed next to it in the next turn?1

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Answered: 1. A game consists of equilateral…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.4: Applications

Problem 26EQ

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2. Suppose that in Example 2.27, 400 units of food A, 500 units of B, and 600 units of C are placed in the test tube each day and the data on daily food consumption by the bacteria (in units per day) are as shown in Table 2.7. How many bacteria of each strain can coexist in the test tube and consume all of the food? Table 2.7 Bacteria Strain I Bacteria Strain II Bacteria Strain III Food A 1 2 0 Food B 2 1 3 Food C 1 1 1
Consider a two-player game that is set up with two piles of stones. The two players are taking turns removing stones from one of the two piles. In each turn, a player must choose a pile and remove one stone or two stones from it. The player who removes the last stone (making both piles empty) wins the game. Show that if the two piles contain the same number n ∈ Z+ of stones initially, then the second player can always guarantee a win.
A three-finger Morra game is a game in which two players simultaneously show one, two, or three fingers at each round. The outcome depends on a predetermined set of rules. Here is an interesting example: If the numbers of fingers shown by A and B differ by 1, then A loses one point. If they differ by more than 1, the round is a draw. If they show the same number of fingers, A wins an amount equal to the sum of the fingers shown. Determine the optimal strategy for each player. (Enter your probabilities as fractions.) Player A should show one finger with probability _____ , two fingers with probability _____ , and three fingers with probability ______ . Player B should show one finger with probability _______ , two fingers with probability ________ , and three fingers with probability ________ . Find the expected value of the game. The expected outcome is the player A will win __________ points per round, on average.
Consider the single object allocation problem discussed in the class. A single object needs to be allocated to one of n agents. Each agent has a value , that is, the utility he derives from the object, if given for free. The game is as follows: ? Each player/agent simultaneously announces a non-negative number - call it his bid. Denote the bid by player i as bi ≥ 0. Highest bidder wins the object - in case of a tie, the bidder with the lowest index wins (for instance, if agents 2, 3, 5 have the highest bid, then 2 wins the object). The winner gets the object for free, i.e., does not pay anything. All other agents ( i.e., those who don’t get the object) receive a payment equal to the highest bid amount. A) Formulate the game in Normal Form. B) Verify whether the game has a weak dominant strategy equilibrium. Explain why, or why not .
SO what would be the L, Lq, and Wq of this problem? Assuming we are trying to develop and sovle a waiting line system that can accomodate this increased leel of passenger traffic.

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