There are \(19\) adventurers standing in a queue to see a dragon’s treasure. They can enter the cave in three groups, with \(15\) minute breaks between two consecutive groups. The order in which adventurers will enter the cave is fixed – they are in a queue after all. But they can still decide who will be in the first, second and third group. Each group has to consist of at least one adventurer. In how many ways can they do that?
Ten players were entered into a badminton tournament. The first round consisted of 5 matches, with each player in one match. In how many different ways could the 10 players be matched against each other?
There are again some adventurers standing in a queue to see a dragon’s treasure. This time, there are more of them – \(26\). The rules have changed slightly, they still enter exactly in the order they are queuing, but they now have to divide themselves into \(5\) groups, and some of the groups can be empty, do not consist of any adventurers at all. In how many ways can they do that now?
Problems often involve a protagonist, a quest and a story. In combinatorics, stories can help us prove identities and formulas, that would be difficult to prove otherwise. Here, you can write your own story, which will show that the following statement is always true:
The number of ways we can choose \(k\) out of \(n\) items is equal to the number of ways we can choose \(k\) out of \(n-1\) objects PLUS the number of ways in which we can choose \(k-1\) out of \(n-1\) objects.
Becky and Rishika play the following game: There are 21 biscuits on the table. Each girl is allowed to take 1, 2 or 3 biscuits at once. The girl who cannot take any more biscuits loses. Rishika starts – show that she can always win.
Alice and Bob play a game, Alice will go first. They have a strip divided into \(2026\) identical squares. In each move, they put a \(2 \times 1\) domino block on the strip, covering two full squares. The person that is not able to make their move loses. Who has a winning strategy?
Ana and Daniel are playing a game that involves a chocolate bar. The top left square of the bar is poisoned. In each move, a player has to pick a square and take all the pieces contained in the rectangle whose top left corner is the selected square and the bottom right corner is the bottom right corner of the whole bar. The person who takes the poisoned square loses. Who has a winning strategy if Daniel starts?
Two pirates are playing a game. They have \(42\) gold coins on a table. Each of them is allowed to take either \(1\) or \(5\) coins from the table. The pirate who takes the last coin wins. Who will win – the first pirate or the second pirate?
Rekha and Misha also play with coins. They have an unlimited supply of 10p coins and a perfectly round table. In each move, one of them places a coin somewhere on that table, but not on top of any other coins already there. A person that cannot place any more coins loses. Who will win, if Rekha goes first?
Varoon and Mahmoud are given two plates of fruit. On one plate, there are \(13\) apples, on the other, there are \(16\) pears. Each of the boys can take any number of fruit from one plate when he moves. The person who takes the last fruit wins. If Mahmoud starts, who will win?