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##### Discrete Mathematics

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**Discrete Mathematics**In Problems Reporter: Anton Kuznietsov, Kharkiv Karazin National University, Ukraine**Discrete mathematics and programming**Ideas from combination theory and graph theory Math packages and programming are applied in are applied in Some problems in discrete mathematics Algorithmic programming**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**a = true A – knight**A: B: 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: 1. Knights & Liars - solution ↯ b at least 2 said truth, ↯ Answer: B is the only knight, A and C are liars.**0 1 1 0**1 1 1 0 1 0 0 0 0 0 0 1 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 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 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 11 1 0 0 0 1 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 2. Competing people - solution A - liar B - liar C - liar D - liar Answer: Charles is a liar, Bill is the champion.**(1)**(2) (3) (4) 3. Search for the culprit A A is guilty • 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?**(1)**(2) (3) (4) 3. Search for the culprit - solution A B C Answer: D is guilty.**4. Queens**• Dispose eight queens on the chess-board so, that the queens don't threaten each other. • Find all variants of such arrangement.**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**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).**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**6. Pavement**2 x N • 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. N = 1, 2, 3, … 1, 4, 9, 25, 64, 169, 441, …**– the number of ways to pave the road.**6. Pavement 2 x N**6. Pavement**2 x N N = 1, 2, 3, … 1, 4, 9, 25, 64, 169, 441, …**6. Pavement**2 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?**6. Pavement**3 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.**- the required quantity;**- the number of ways to pave this road: 6. Pavement 3 x N , m = 1, 2, 3, … Am = 3, 11, 41, 153, 571, 2131, 7953, …**Conclusions**Combination theory, graph theory, pounding theory, Fibonacci numbers, Catalan numbers Programming are applied in are applied in Algorithmic programming Problems of logic, combination theory, graph theory**Thank you for your kind attention!**Reporter: Anton Kuznietsov, Kharkiv Karazin National University, Ukraine