Problems

a) While visiting Cape Verde, Pirate Jim and Pirate Bob bought several chocolate chip cookies each. Jim paid 93 copper coins for his cookies and Bob paid 102 copper coins. What could be the price of a single cookie if it is a natural number?

b) Captain Hook and Captain Kid bought several tricorn hats each. Captain Hook paid 6 silver coins more than Captain Kid. What could be the price of a tricorn hat if it is an integer?

Show that $a-b$ is divisible by the greatest common divisor of $a$ and $b$.

a) Can you measure $6$ litres of water using two buckets of volumes $7$ and $10$ litres respectively?

b) Can you measure $7$ litres of water using buckets of volumes $9$ and $12$ litres respectively?

a) King Haggard has a velvet pouch filled with diamonds. He can divide these diamonds into 3 equal piles, 4 equal piles, or 5 equal piles. How many diamonds does he have if it is known that his collection contains less than 100 diamonds in total?

b) King Haggard has a stash of gold coins. He is one coin short of being able to divide these coins into 4 equal piles, or 5 equal piles, or 6 equal piles, or 7 equal piles. How many coins does he have if he has fewer than 500?

Suppose that $p$ is a prime number.

a) How many numbers that are less than $p$ are relatively prime to it?

b) How many numbers that are less than $p^2$ are relatively prime to it?

I have written $5$ non-prime numbers on a piece of paper and hidden it in a safe locker. Every pair of these numbers is relatively prime. Show that at least one of these numbers has to be larger than a $100$.

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 $2018$ identical squares. In one move, they put a $2\times 1$ domino block on the strip, covering two full squares. One that is not able to make their move, loses. Who has the 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 will win, 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?