COMP 2711H: Lecture 5

Date: 2024-09-09 17:57:10

Reviewed:

Topic / Chapter: Practice Problems

summary

❓Questions

Notes

Even More Problems

Problem 1
- prison guard
- prisoner $n, m$ gets to assigned a number ( $n, m$ )
  - they know what number they were assigned
  - and $∣ n - m ∣ = 1$
- can say: 1 sentence, only once
  - "I know out number and it's (n,m)"
- show that: they will be able to come to an answer
- $i := min (n, m)$ :
- base case: $i = 0$
  - one with 0 can immediately shout out $(n, m) = (0, 1)$
  - solved on day 1
- base case: $i = 1$
  - if either number was 0, the prisoner would have said $(n, m) = (0, 1)$
  - however, if no one shouted it on day 1, prisoner w/ no. 1 can know that his/her number is the min. and the other no. is $i + 1 = 2$
  - solved on day 2
- $p (i)$ : if $i = min (n, m)$ , then it will be announced on day $n + 1$
- $\forall i \in N, p (i) ⟹ p (i + 1)$
  - as, if it was $p (i)$ , then it would have been announced on day $i + 1$
  - it wasn't the case, so the prisoner with $i + 1$ can announce $(n, m) = (i + 1, i + 2)$ on day $i + 2$
Problem 1'
- a prisoner: supposed to be executed
- someday next week, but not known
  - 👨‍🏫 and you will be surprised
- prisoner: cannot be executed on Sunday
  - as he won't be surprised on that day
- prisoner: cannot be executed on Saturday
  - as he won't be surprised to be so by the end of Friday
- and so on...
- prisoner: I ought not to be executed!
- answer: he was executed on Wednesday, and he was surprised
Problem 2
- 2d lattice: all points of form $(x, y)$ where $x, y \in Z$
- for each point: has natural number written on it
- number written on point: will be mean of its neighbors
  - neighbor: i.e. on m
- prover all the number are equal
- proof by well ordering
  - $X =$ all numbers assigned to points
    - $X \subseteq N \land X \neq = \emptyset$
    - $X$ has a minimum value
  - and the minimum value's neighbors: must all be $m$
    - unless: there must be a neighbor that is bigger than $m$
    - then there also must be a neighbor with natural number smaller than $m$ , too
Problem 3
- there is a circle w/ circumference 1
- and there are $n$ cars on its circumference
  - and each can go distance $x_{i}$ , and fuel can be shared
- if $\sum_{i = 1}^{n} x_{i} \geq \sum_{i = 1}^{n} d_{i} = 1$ , there exists a car that can catch & steal others' fuel, and make a full round
- base case: $n = 1$
  - with only 1 car, w/ $x_{i} > 1$ , then the car can make a whole round
  - or: $\sum_{i = 1}^{1} x_{i} \geq \sum_{i = 1}^{1} d_{i}$
- induction case: $p (n) \to p (n + 1)$
  - $d_{i}$ : denotes distance till the next car
  - there exists at least 1, "good" car that can catch another car ( $d_{i} \leq x_{i}$ )
    - however, it doesn't mean that the car can make a whole round after the catch
  - 👨‍🏫 idea: removing the "good" car, and making it with $n$ cars only
    - remove the good car $x_{i}$ , and eliminate $d_{i}$
    - and change fuel of car $i + 1$ to $x_{i + 1} + x_{i} - d_{i}$
    - intuition: if a car could reach car $i$ , then it can reach car $i + 1$ with this wormhole & fuel change
      - 👨‍🎓 it is, kind of equivalent, considering car $i$ is a "good" car
    - and, still, this subproblem w/ $n$ cars: satisfy $\sum_{i = 1}^{n} x_{i} \geq \sum_{i = 1}^{n} d_{i}$
- ⭐👨‍🏫 we don't get to choose the instance of $n + 1$ cars, as we must show it holds for all combinations
  - however, we can do so for $n$ cars, as it is the assumption we are having
Problem 4
- there are no four positive integers $x, y, z, u$ s.t.
  - $x^{2} + y^{2} = 3 (z^{2} + u^{2})$
- proof by contradiction
  - assume: assignment s.t. making $(x^{2} + y^{2})$ minimum exists
    - $(x, y, z, u) = (a, b, c, d)$
  - $a^{2} + b^{2} = 3 (c^{2} + d^{2})$
    - RHS: multiple of 3
    - LHS: must also be multiple of 3
      - $a \equiv_{3} 0 ⟹ a^{2} \equiv_{3} 0$
      - $a \equiv_{3} 1 ⟹ a^{2} \equiv_{3} 1$
      - $a \equiv_{3} 2 ⟹ a^{2} \equiv_{3} 1$
    - and sum of two such numbers: must be within ${0, 1, 2}$
      - and $3∣ a^{2} + b^{2}$ only occurs when both numbers are multiple of 3
    - if $3∣ a^{2} ⟹ 3∣ a$
      - $a = 3 a^{'}$
    - if $3∣ b^{2} ⟹ 3∣ b$
      - $b = 3 b^{'}$ $a^{2} + b^{2} 9 a^{'2} + 9 b^{'2} 3 a^{'2} + 3 b^{'2} 3 (a^{'2} + b^{'2}) c^{2} + d^{2} = 3 (c^{2} + d^{2}) = 3 (c^{2} + d^{2}) = c^{2} + d^{2} = c^{2} + d^{2} = 3 (a^{'2} + b^{'2})$
    - and $c^{2} + d^{2}$ : smaller then $a^{2} + b^{2}$ , clearly
    - contradiction: thus no such solution exists
Problem 5
- graph problem: a country w/ $n$ cities
- and exactly 1 road between any two cities
  - and all are 1-way
- a city $C$ : a central city
  - if every other city $C^{'}$ can reach $C$ directly / or with 1 stop
- show that there exists at least 1 central city
- proof by induction
  - $n = 1, 2$ : it works
  - for $n$ city: all other cities will have road to
    - $c$
    - or other city that has road to $c$
  - by adding a new city $c_{n + 1}$
    - if $c_{n + 1}$ has road to $c$ : $c$ remains the central
    - if $c$ has road to $c_{n + 1}$ :
      - if $c_{n + 1}$ has any road to $c$ 's first neighbor: $c$ remains the central
      - else:
        
        $c$ and first neighbors of $c$ : has direct road to $c_{n + 1}$
        
        second neighbors of $c$ : can access $c_{n + 1}$ through first neighbors
        
        $c_{n + 1}$ is the new central
- extremal proof
  - count: how many roads are coming in
    - such count: $\in [0, n - 1]$
    - i.e. indegree
  - if $c$ : a city w/ maximal indegree
    - then $c$ is the capital
  - proof
    - there are first neighbor cities of $c$
    - and other cities
      - if it has road to first neighbors: can access $c$ in two steps
        
        doesn't make $c$ not-a-capital
    - $c$ is not capital only when:
      - $c$ and all first neighbors are towards a different city
        
        👨‍🎓 as: if that city had road towards any of first neighbors / $c$ , it can reach $c$ easily
        
        must be the other direction
        
        i.e. if $c$ had $k$ indegree, new city has at least $k + 1$ indegree
      - however, it's a contradiction in choice of $c$
        
        as $c$ is supposed to be a city w/ maximal indegree

COMP 2711H: Honors Discrete Mathematical Tools for Computer Science