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Discrete Mathematics Reporter: Anton Kuznietsov, Kharkiv Karazin National University, Ukraine In Problems

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Discrete mathematics and programming Ideas from combination theory and graph theory Algorithmic programming Some problems in discrete mathematics Math packages and programming are applied in

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Scheme of the presentation Problems 1. Knights and Liars 2. Competing people 3. Search for the culprit 4. Queens 5. Knight’s move 6. Pavement Conclusions

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1. Knights & Liars Suppose, we are on a certain island and have talked with three inhabitants A, B and C. Each of them is either a knight or a liar. Knights always say truth, liars always lie. Two of them (A and B) came out with the following suggestions: A: We all are liars. B: Exactly one of us is a knight. Question: Who of the inhabitants A, B and C is a knight, and who is a liar? Write down the inhabitants’ propositions, using formulas of proposition calculus. a = true A – knight A:B:

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1. Knights & Liars - solution a = true A – knight A:B: at least 2 said truth, ↯ ↯ b Answer: B is the only knight, A and C are liars.

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2. Competing people Four boys – Alex, Bill, Charles and Daniel – had a running-competition. Next day they were asked: “Who and what place has taken?” The boys answered so: Alex: I wasn’t the first and the last. Bill: I wasn’t the last. Charles: I was the first. Daniel: I was the last. It is known, than three of these answers are true and one is false. Question: Who has told a lie? Who is the champion? 0 1 1 0 1 1 1 0 1 0 0 0 0 0 0 1

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2. Competing people - solution 0 1 1 0 1 1 1 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 1 1 0 0 1 1 1 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 1 A - liarB - liarC - liarD - liar 0 1 1 0 1 1 1 0 0 1 1 1 0 0 0 1 0 1 1 0 1 1 1 0 0 1 1 1 0 0 0 1 Answer: Charles is a liar, Bill is the champion. 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 0 1 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 0 0 0 0 0 0 1

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3. Search for the culprit Four people (A, B, C, D) are under suspicion of committing a crime. The following is ascertained: If A and B are guilty, then the suspected C is also guilty. If A is guilty, then B or C is also guilty. If C is the culprit, then D is also guilty. If A is innocent, then D is the culprit. Question: Is D guilty? A A is guilty (1) (2) (3) (4)

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3. Search for the culprit - solution (1) (2) (3) (4) A B C Answer: D is guilty.

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4. Queens Dispose eight queens on the chess-board so, that the queens don't threaten each other. Find all variants of such arrangement.

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4. Queens - solution 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0

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5. Knight’s moves There is a chess-board of size n x n (n <= 10). A knight stands initially on the field with coordinates (x0, y0). The knight has to visit every field of the chess-board exactly once. Find the sequence of knight’s moves (if it exists).

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5. Knight’s moves - solution 1 10 5 16 25 4 17 2 11 6 9 20 13 24 15 18 3 22 7 12 21 8 19 14 23

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6. Pavement Roadmen have pavement plates of size 1x1 and 1x2. How many ways are there to pave the road of size 2xN (1<=N<=1000)? The plates 1x2 are made on factory so, that they can be placed only with the wide side lengthwise the road. 2 x N 1, 4, 9, 25, 64, 169, 441, … N = 1, 2, 3, …

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6. Pavement2 x N – the number of ways to pave the road.

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1, 4, 9, 25, 64, 169, 441, … 6. Pavement2 x N N = 1, 2, 3, …

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6. Pavement Roadmen have only plates of size 1x2. The plates can be placed both lengthwise and crosswise the road. How many ways are there in this case? 2 x N

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6. Pavement3 x N Roadmen have only plates of size 1x2. The plates can be placed both lengthwise and crosswise the road. How many ways are there in this case? 1 < N < 1000. N is even.

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6. Pavement3 x N, - the required quantity; - the number of ways to pave this road: A m = 3, 11, 41, 153, 571, 2131, 7953, … m = 1, 2, 3, …

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Conclusions Combination theory, graph theory, pounding theory, Fibonacci numbers, Catalan numbers Algorithmic programming Problems of logic, combination theory, graph theory Programming are applied in

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Thank you for your kind attention! Reporter: Anton Kuznietsov, Kharkiv Karazin National University, Ukraine

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