Consider a sequence of real numbers (assume is even). We are interested in partitioning the numbers into pairs (that is, each of the numbers appears in exactly one of the pairs) in the following way. For each pair in the partition, compute the sum of its numbers. Denote by the sums. The goal is to find the partition that minimizes the maximum sum. (a) Describe concisely in English the idea of a brute-force algorithm for solving this problem, and provide, using big- notation, the worst-case running time of your algorithm. (b) Establish a mathematical result that will allow you to design a worst-case polynomial-time algorithm for the problem, and prove the result. (c) Describe the polynomial-time algorithm and provide, using big- notation, its worst-case running time.

Program using c++ only on visual studio pig is a simple two player dice game, played with one die. the first player to reach or surpass 50 is the winner. each player takes a turn rolling the dice. they add to the pot with each roll, having to decide to roll again and increase the pot, or cash out. the risk being they could lose the amount they’ve accumulated into the pot. the rules for each player’s die roll. 1. roll the dice. a. if user rolled a 1, i. the pot gets set to zero ii. the other player goes to step 1. b. a roll of 2-6 is added to the pot. 2. user can choose to hold or roll again. a. choice roll. return to step 1. b. choice hold. i. increment player score by the pot amount. ii. pot gets set to 0. iii. second player gets to roll and goes to step 1. program requirements: ● before each opponent begins ○ output the score for the person and the computer. ○ output the opponents whose turn is beginning and ask the user to hit enter to continue. ● with each dice roll. ○ output the die value, and amount of the round pot. ○ if it’s the users roll ask if they want to roll again ( r ) or hold ( h ). your program should allow r, r, h or h as valid input. if input is anything else, ask the user again until valid input is obtained. ○ the ai will continue playing until the round pot is 20 or more. ● once a player’s score is greater or equal to 50 then they have won, it will no longer ask if they want to keep rolling the die or not. ● once there is a winner ○ score totals are output along with who the winner was. user or computer ○ player is asked if they want to play again y or n. valid input should be y, y, or n, n. ● when a new game starts the starting roll goes to the player that did not roll last. if the user rolled last in the previous game, then the computer rolls first and vice versa. when the program first begins, the player will make the first roll of the first game. development notes : ● you will need a way to roll dice in your program. the rand() function works well, but returns an integer. if we want numbers 0 – 9 we can get the value modulus 10. ● call srand() with a value to seed it. it’s common to seed it with the current computer clock, include ctime, and then call srand(time(

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:you areto design the controller for alight that functions both as an ordinary light and also as a motion activated light and alarm. a.if the manual switch s is on, then the light l is on. b.besides the manual switch, there is a motion detector, m1, which activatesthis light.c.if motion is detected but the light is on anyway because s is on, only then a secondoutput a, an alarm, is turned on. d.the disable switch, d, disables the motion activated light and alarmbut leaves manual control operation of the light using switch s.(i)read the problem statement and clearly identify the inputs and outputs for the circuit you are designing. (ii)create the truth table for this system; include the light, alarm, switch, disable, and the motion sensor.(iii)draw a schematic of this system.