3. (10 pts.) You are in charge of the United States Mint. The money-printing machine has developed a strange bug: it will only print a bill if you give it one first. If you give it a d-dollar bill, it is only willing to print bills of value d² mod 400 and d² + 1 mod 400. For example, if you give it a $6 bill, it is willing to print $36 and $37 bills, and if you then give it a $36-dollar bill, it is willing to print $96 and $97. You start out with only a $1 bill to give the machine. Every time the machine prints a bill, you are allowed togive that bill back to the machine, and it will print new bills according to the rule described above. You want to know if there is a sequence of actions that will allow you to print a $20 bill, starting from your $1 bill.Model this task as a graph problem: give a precise definition of the graph (what are the vertices and edges)involved and state the specific question about this graph that needs to be answered. Give an algorithm to solve the stated problem and give the running time of your algorithm.

[10 pts] Suppose John's biking environment consists of n ≥ 3 landmarks, which are linked by bike route in a cyclical manner. That is, there is a bike route between landmark 1 and 2, between landmark 2 and 3, and so on until we link landmark n back to landmark 1. In the center of these is a mountain which has a bike route to every single landmark. Besides these, there are no other bike routes in the biking environment. You can think of the landmarks and the single mountain as nodes, and the bike routes as edges, which altogether form a graph G. A path is a sequence of bike routes. (a) [6 pts] Find the number of paths of length 2 in the graph in terms of n. Justify your answer. (b) [6 pts] Find the number of cycles of length 3 in the graph in terms of n. Justify your answer. (c) [6 pts] Find the number of cycles in the graph in terms of n. Justify your answer.

3. The coating experiment, described in Excercise 7 of Chapter 7, was to study the effect of different spray parameters on thermal spray coating properties. In the experiment, the authors attempted to produce high-quality alumina (Al2O3) coatings by controlling the fuel ratio (factor A at 1:2.8 and 1:2.0), carrier gas flow rate (factor B at 1.33 and 3.21 L s-¹), frequency of detonations (factor C at 2 and 4 Hz), and spray distance (factor D at 180 and 220 mm). To quantify the quality of the coating, the researchers measured multiple response variables. In this example we will examine the porosity (vol. %). The data are shown in the table below and can be downloaded from http: //deanvossdraguljic.ietsandbox.net/DeanVossDraguljic/SAS-data.html. A B с D Yijkl A B C D 2 2 2 2 5.95 2 2 2 1 2 2 1 2 1 4.57 4.03 2.17 1 2 2 2 1 2 2 2 2 1 1 1 2 1 2 1 1 1 1 1 2 1 1 1 1 2 2 1 2 1 2 1 2 Yijkl 12.28 9.57 6.73 6.07 8.49 4.92 6.95 5.31 1 2 2 3.43 2 1 2 1 1.02 2 1 1 2 4.25 2 1 1 1 2.13 1 1 1 1 (a). [3pts]. Run a model with ALL main-effects and two-way interaction effects. Write down the SAS code copy the ANOVA table from SAS. and (b).[2pts] The 95% confidence interval for the difference between the two main effects of B is ( (c).[2pts] Do you believe there are significant interaction effects between B and C? Answer yes or no. (d).[3pts] Generate an interaction plot for the B*C interaction. State how the plot supports your answer in c. [Hint: use 1smeans B*C to get the least squares estimates, then plot it using any software of your choice.] (e).[3pts] The 90% confidence upper limit for the error variance o² is Show your work.

2. Consider the reaction time experiment described in Exercise 4 of Chapter 4. a. [3pts] Write down the two-way complete model for the experiment. Remember to explain each term in the model. b. [3pts] Find the sums of squares that are accounted for by the factors and their interactions, i.e., ssA, ssB and ssAB. c.[5pts] Generate an interaction plot. Do you see an obvious interaction between the two factors? Carry out a formal hypothesis testing for the interaction. d. [3pts] Test the hypothesis that different elapsed times have the same effects on the reaction time. e.[3pts] Find a 95% confidence interval for the difference between the average reaction time from the auditory cue and the average reaction time from the visual cue [Hint: this is to compare the two main effects of factor cue]. f.[3pts] Find an appropriate confidence interval for the difference between auditory cue and the visual cue, when the elapse time is 5 seconds.

Q3 (10 points) Let S and A be two finite nonempty sets of integers. Define a digraph D with vertices V(D)= A, where (x, y) is an arc of D if z ‡y and y-ze S. 1. Draw the digraph D for A = {0, 1, 2, 3, 4) and S = (-2,1,2,4). 2. What can be said about Dif A and S consist only of odd integers? 3. If |A|-|S-5. how large can the size of D be?

1. You have been hired as a consultant by the company "Bird Baths R Us” to help determine why one of their products is frequently returned by customers.The CEO expects you to justify your conclusions using both graphical and numerical data. You should aim to be as precise as possible in your analysis.Please upload screenshots of any graphical material as well as any Excel files (not screenshots) you use. The product in question is a hemispherical bird bath known as the "Avosphere."It is ten inches deep and features six perches for visiting birds. The height x, in inches, of the water as the Avosphere is being filled is modeled by the differential equation: \frac{d x}{d t}=\frac{60\left(1-20 k x+k x^{2}\right)}{20 x-x^{2}} where the time t is measured in hours and k is a constant that measures how quickly water evaporates. If there were no evaporation, k would be zero. Your preliminary tests have determined k to be .02. We will assume the bird bath initially has 1 inch of water. • In the "DFIELD Direction Field" window menu bar, select Options → Delay Time Per Point → 10 Milliseconds Options → Solution Direction → Forward ● In the “DFIELD Equation" window, you can change the values in the"Display Window." Use Min t = Min x = 0. You will need to decide what you want the maximum values to be. (a) Using your evaporation constant, what is the height of the water after 2hours? What would the height be after 2 hours if there were no evaporation? (b) Assuming no evaporation, how long until the bird bath is full of water? (d) Based on your answers above, why do you think customers are dissatisfied with this product? (e) Approximately what value would Bird Baths R Us need to reduce theevaporation constant to so that customers can get the Avosphere at least70% full? (c) With your evaporation constant, what is the maximum depth of water acustomer can achieve in their Avosphere?

Task Poker machines are devices specially designed to take money from gamblers. They seduce players with bright lights, lots of sounds and enough wins, including the lure of very rare jackpots, to keep players interested. The mandated minimum 'return to player' for machines in South Australia is 87.5% of the money gambled (Office for Problem Gambling, n.d.). Or, put another way, for every $100 gambled, the player can expect, in the long run, to lose $12.50. However, all is not what it seems when it comes to poker machine maths. A gambler could lose all the money they put into a machine yet would still be considered to have received 87.5% of the amount they gambled even though they walk away with nothing. This last statement probably doesn't make much sense at first glance. The key to the explanation is that the 'return to player' is calculated using the money gambled - not what is actually spent by a player. Refer to the ABC article 'Poker Machine Maths.

3. Consider the following network of computers: E 3 The numbers labelling each edge indicate the number of Macquariecoin it costs to send a message along that network connection. We wish to minimise the costs of sending messages by finding paths of minimal length in this weighted graph. Use Dijkstra's algorithm to construct a shortest-path spanning tree rooted at computer C. In your an- swer, you should: (a) [3 marks]. Draw the graph and illustrate the spanning tree you construct within it. (b) [5 marks]. Show your working by displaying, at each step of execution, the contents of the fringe list, clearly identifying the edge which will be added to the tree next. (c) [3 marks]. List the length of each shortest path starting from the chosen root computer C to each of the other computers in the network.

2. Assume that P = (((p1^-P2) AP3) ⇒ P2). Write down the truth table for P. How many different interpretations make P true. Give the list of interpretations I under which P false. 3. A formula P is called satisfiable if there exists an interpretation I such that I(P) = 1. Which of the following formulas are satisfiable? Check using truth tables. \text { (a) }\left(\neg p_{1} \wedge p_{2}\right) \text { (c) }\left(p_{1} \Leftrightarrow \neg p_{1}\right) \text { (d) }\left(p_{1} \wedge\left(\neg p_{2} \vee \neg p_{1}\right)\right)

Activity #1: Go Moves - team (https://en.wikipedia.org/wiki/Go_(game)) The purpose of this activity is to get you used to using lists of lists, in a 2-D matrix-like format. Your team will create a program that sets up a small Go board and lets users place stones.