Problems

You have in your hands a royal flush! That is, Ace, King, Queen, Jack and $10$ of spades. How many shuffles of your hand swap the Ace and Jack?

In the diagram below, I wish to write the numbers $6,11,19,23,25,27$ and $29$ in the squares, but I want the sum of the numbers in the horizontal row to equal the sum of the numbers in the vertical column. What number should I put in the blue square with the question mark?

Let $\varphi (n)$ be Euler’s function. Namely $\varphi (n)$ counts how many integers from $1$ to $n$ inclusive are coprime with $n$. For two natural numbers $m$, $n$ such that $gcd(m,n)=1$, prove that $\varphi (mn)=\varphi (m)\varphi (n)$.

For any positive integer $k$, the factorial $k!$ is defined as a product of all integers between 1 and $k$ inclusive: $k!=k\times (k-1)\times ...\times 1$. What’s the remainder when $2025!+2024!+2023!+...+3!+2!+1!$ is divided by $8$?

There are two imposters and seven crewmates on the rocket ‘Plus’. How many ways are there for the nine people to split into three groups of three, such that each group has at least two crewmates? The two imposters and seven crewmates are all distinguishable from each other, but we’re not concerned with the order of the three groups.

For example: $\{I_1,C_1,C_2\}$, $\{I_2,C_3,C_4\}$ and $\{C_5,C_6,C_7\}$ is the same as $\{C_3,C_4,I_2\}$, $\{C_5,C_6,C_7\}$ and $\{I_1,C_2,C_1\}$ but different from $\{I_2,C_1,C_2\}$, $\{I_1,C_3,C_4\}$ and $\{C_5,C_6,C_7\}$.

Draw the plane tiling with regular hexagons.

A gang of three jewel thieves has stolen some gold coins and wants to divide them fairly. However, they each have one unusual rule: (i) The first thief wants the number of coins to be divisible by $3$ so they can split it evenly. (ii) The second thief wants the number of coins to be divisible by $5$ because she wants to split her share with her four siblings. (iii) The third thief wants the number of coins to be divisible by $7$ since he wants to split his share amongst seven company stocks.

However, they’re stuck as the number of coins isn’t divisible by any of these numbers. In fact, the number of coins is $1$ more than a multiple of $3$, $3$ more than a multiple of $5$ and $5$ more than a multiple of $7$.
What’s the smallest number of coins they could have? (And if you’re feeling generous, how would you help them out?)

Munira wants to put $6$ books on her shelf, $4$ of which are red and $2$ of which are blue. The four red ones are a small paperback, a small hardback, a large paperback and a large hardback. The two blue ones are both paperback, one small and one large. She doesn’t want the two blue ones next to each other. In how many ways can she do this?

Imagine a cube that’s turquoise on the front, pink on top, yellow on the right, white on left, dark blue on back and orange on the bottom. If Arne rotates this ${180}^{\circ }$ about the line through the middles of the turquoise and dark blue sides, then does it again, he gets back to the original cube. If Arne rotates this ${90}^{\circ }$ about that same line, then does that three more times, then he also gets back to the original cube.
Is there a rotation he could do, and then do twice more, to get back to the original cube?

Can you tile the plane with regular octagons?