A new coin that is worth exactly 4 cents has been add to our currency system. Prove or disprove the statement: "The cashier`s algorithm using quarters, dimes, nickels, 4 cent coins, and pennies and will always produce change using the fewest coins possible
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A new coin that is worth exactly 4 cents has been add to our currency system. Prove or disprove the statement: "The cashier`s
pennies and will always produce change using the fewest coins possible."
Allow us first to demonstrate unequivocally that the coin - changing issue has ideal foundation.
Assume we have an ideal answer for an issue of making change for n pennies, and we realize that this ideal arrangement utilizes a coin whose worth is c pennies; let this ideal arrangement utilize k coins. We guarantee that this ideal answer for the issue of n pennies should hold inside it an ideal answer for the issue of n - c pennies. We utilize the standard reorder contention. Obviously there are k - 1 coins in the answer for the n - c pennies issue utilized inside our ideal answer for the n pennies issue. Assuming we had an answer for the n - c pennies issue that utilized less than k - 1coins, then, at that point, we could utilize this answer for produce an answer for the n pennies issue that utilizes less than k coins, which goes against the optimality of our answer.
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- Consider a group of balls where each ball is one of k colors. You can assume that there is an equal number of balls of each color. Each ball, aside from its color, has a written integer on it. The numbers are in no way associated with the color. a) Devise an algorithm that orders all the balls according to colors in a way that no ball comes before a ball of a lighter color. In other words, how do you order balls from lightest to darkest, however, inside one group (i. e., one color), balls can be ordered in any way with respect to the number written on the ball. Design an algorithm and analyze it. b) Now take the output from (a), and also order balls within the color. How much time is needed for this step?Correct answer will be upvoted else Multiple Downvoted. Don't submit random answer. Computer science. You are given a variety of n integers a1, a2, ..., an, and a set b of k unmistakable integers from 1 to n. In one activity, you might pick two integers I and x (1≤i≤n, x can be any integer) and allocate ai:=x. This activity should be possible provided that I doesn't have a place with the set b. Compute the base number of tasks you ought to perform so the cluster an is expanding (that is, a1<a2<a3<⋯<an), or report that it is inconceivable. Input The principal line contains two integers n and k (1≤n≤5⋅105, 0≤k≤n) — the size of the exhibit an and the set b, individually. The subsequent line contains n integers a1, a2, ..., an (1≤ai≤109). Then, at that point, if k≠0, the third line follows, containing k integers b1, b2, ..., bk (1≤b1<b2<⋯<bk≤n). On the off chance that k=0, this line is skipped. Output In case it is difficult to make the exhibit…Correct answer will be upvoted else downvoted. Computer science. You are given a positive (more prominent than nothing) integer n. You need to address n as the amount of integers (perhaps negative) comprising just of ones (digits '1'). For instance, 24=11+11+1+1 and 102=111−11+1+1. Among every single imaginable portrayal, you need to track down the one that utilizes the base number of ones altogether. Input The single line contains one integer n (1≤n<1050). Output Print one integer x — the base number of ones, with the end goal that there exist a portrayal of n as the amount of integers (potentially bad) that utilizes x ones altogether.
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- Correct answer will be upvoted else Multiple Downvoted. Don't submit random answer. Computer science. Sasha likes exploring diverse mathematical articles, for instance, wizardry squares. However, Sasha comprehends that enchanted squares have as of now been examined by many individuals, so he sees no feeling of concentrating on them further. All things considered, he designed his own kind of square — a superb square. A square of size n×n is called prime if the accompanying three conditions are held all the while: all numbers on the square are non-negative integers not surpassing 105; there are no indivisible numbers in the square; amounts of integers in each line and every segment are indivisible numbers. Sasha has an integer n. He requests you to view as any great square from size n×n. Sasha is certain beyond a shadow of a doubt such squares exist, so help him! Input The principal line contains a solitary integer t (1≤t≤10) — the number of experiments. Every one…Correct answer will be upvoted else Multiple Downvoted. Computer science. Athenaeus has recently wrapped up making his most recent melodic piece and will introduce it tomorrow to individuals of Athens. Tragically, the tune is somewhat dull and almost certain will not be met with a warm gathering. His tune comprises of n notes, which we will treat as certain integers. The variety of a tune is the number of various notes it contains. As a supporter of music, Euterpe looks after arrangers and guides them all through the method involved with making new tunes. She chose to help Athenaeus by changing his melody to make it more assorted. Being a minor goddess, she can't self-assertively change the tune. All things being equal, for every one of the n notes in the tune, she can either leave it for what it's worth or increment it by 1. Given the tune as an arrangement of integers portraying the notes, discover the maximal, attainable variety. Input The input comprises of numerous…Find the error in the following argument: Theorem. All computer programs contain the same number of bugs. Proof. If we show that any set of n programs contains the same number of bugs, then we have proved the theorem. We argue by induction on n. If n=1 then in any set of 1 program, all the programs contain the same number of bugs, so the statement is true, in this case. Now suppose that for every set of programs containing less then n programs, all the programs have the same number of bugs. Let DD be a set of n programs p1,p2,…,pn. Now the set D1={p2,…,pn} contains n−1 programs and hence they all contain the same number of bugs, by the induction hypothesis. Similarly the set D2={p1,…,pn−1} contains n−1 programs and hence they all contain the same number of bugs. In particular p1 and pn contains the same number of bugs as the other programs in the set. Hence all the programs contain the same number of bugs.
- Computer science. Correct answer will be upvoted else downvoted. You have an at first void cauldron, and you need to blend an elixir in it. The elixir comprises of two fixings: enchantment pith and water. The elixir you need to blend ought to contain precisely k % sorcery substance and (100−k) % water. In one stage, you can pour possibly one liter of sorcery pith or one liter of water into the cauldron. What is the base number of steps to mix a mixture? You couldn't care less with regards to the complete volume of the elixir, just with regards to the proportion between sorcery substance and water in it. A little update: in the event that you pour e liters of embodiment and w liters of water (e+w>0) into the cauldron, then, at that point, it contains ee+w⋅100 % (without adjusting) sorcery substance and we+w⋅100 % water. Input The primary line contains the single t (1≤t≤100) — the number of experiments. The sole line of each experiment contains a solitary integer k…Correct answer will be upvoted else Multiple Downvoted. Computer science. You are given a positive number x. Observe the littlest positive integer number that has the amount of digits equivalent to x and all digits are unmistakable (extraordinary). Input The principal line contains a solitary positive integer t (1≤t≤50) — the number of experiments in the test. Then, at that point, t experiments follow. Each experiment comprises of a solitary integer number x (1≤x≤50). Output Output t replies to the experiments: on the off chance that a positive integer number with the amount of digits equivalent to x and all digits are diverse exists, print the littlest such number; in any case print - 1.There are n different sizes of boxes, from 1 to n. There is an unlimited supply of boxes of each size t, each with a value vt . A box of size t can hold several smaller boxes of sizes a1, a2, . . . , ak as long as the sum of sizes a1 + a2 + . . . + ak is strictly less than t. Each of these boxes may be filled with yet more boxes, and so on. Design an algorithm which runs in O(n2 ) time and finds the maximum value that can be attained by taking one box, potentially with smaller boxes nested inside it.