There are all the numbers from \(1\) to \(2020\) written on the board. Karen and Leon are playing a game where they pick a number off the board and wipe it, together with all of its divisors. Leon goes first - prove that Karen always loses.
Katie and Juan played chess for some time and they got bored - Katie was winning all the time. She decided to make the game easier for Juan and changed the rules a bit. Now, each player makes two usual chess moves at once, and then the other player does the same. (Rules for checks and check-mates are modified accordingly). In the new game, Juan will start first. Show that Katie definitely does not have a winning strategy.
Two players are emptying two drawers full of socks. One drawer has 20 socks and the other has 34 socks. Each player can take any number of socks from one drawer. A player who can’t make a move loses. Who will win, the first or the second player?
Tommy and Claire are going to get some number of game tokens tomorrow. They are planning to play a game: each player can take \(1,4\) or \(5\) tokens from the total. The person who can’t take any more loses. Claire will start. They don’t know how many tokens they will get. They might get a number between \(1\) and \(2020\). In how many cases Claire will win?
Fred and Johnny have a number \(1000\) written on the board. In their turn, a player wipes out a number currently on the board and replaces it with either a number \(1\) smaller, or half of the number on the board (rounded down). A player that writes \(0\) on the board wins. Johnny starts, who will win?
You take nine cards out of a standard deck (ace through 9 of hearts), put them all face up on a table and play the following game against another player:
Both players take turns choosing a card. The first player to have three cards that add up to 15 wins. The ace counts as one.
If both players play optimally, which player has a winning strategy?
Andy and Melissa are playing a game using a rectangular chocolate bar made of identical square pieces arranged in \(50\) rows and \(20\) columns. A move is to divide the bar into two parts along the division line. Two parts of the bar stay in the game as separate pieces and cannot be rotated, but both can continue to be divided. However, Melissa can only cut along the vertical lines and Andy can only cut along the horizontal lines. Melissa starts. Who will win?
Terry and Janet are playing a game with stones. There are two piles of stones, one has \(m\) stones and the other has \(n\) stones initially. In their turn, a player takes from one pile a positive number of stones that is a multiple of the number of stones in the other pile at that moment. The player who cleans up one of the piles wins. Terry starts - who will win?
Mathematical Induction is a method to prove statements that are usually true for all natural numbers. The method consists of two steps.
The first step, known as the base case, is to prove the given statement for the first natural number.
The second step, known as the inductive step, is to prove that the true statement for the number \(n\) implies that the statement for \(n+1\) is also true.
To understand how the method of induction works we look at dominoes. Have you ever seen a line of dominoes falling? How does it happen?
To prove that a line of dominoes will all fall when we push the first one, we just have to prove that:
The first domino falls down (base case)
The dominoes are close enough that each domino will knock over the next one when it falls (inductive step).
Let’s look at triangular numbers, numbers which are a sum of the first \(n\) natural numbers: \[1+2+3+\dots +n\] Show using induction that the \(n\)-th triangular number is equal to \(\frac{n(n+1)}2\).