Sunday Afternoon Maths XXVIII

Here's this week's collection. Answers & extensions tomorrow. Why not discuss the problems on Twitter using #SundayAfternoonMaths or on Reddit.

2009

Source: Teaching Further Maths blog

2009 unit cubes are glued together to form a cuboid. A pack, containing 2009 stickers, is opened, and there are enough stickers to place 1 sticker on each exposed face of each unit cube.

How many stickers from the pack are left?

3$n$+1

Source: Twenty-Six Years of Problem Solving compiled by John Mason

Let $S=\{3n+1:n\in\mathbb{N}\}$ be the set of numbers one more than a multiple of three.

(i) Show that $S$ is closed under multiplication.

ie. Show that if $a,b\in S$ then $a\times b\in S$.

Let $p\in S$ be irreducible if $p\not=1$ and the only factors of $p$ in $S$ are $1$ and $p$. (This is equivalent to the most commonly given definition of prime.)

(ii) Can each number in $S$ be uniquely factorised into irreducibles?

The Ace of Spades

Source: The Week-End Problems by Hubert Phillips

I have three packs of playing cards with identical backs. Call the packs A, B and C.

I draw a random card from pack A and shuffle it into pack B.

I now turn up the top card of pack A, revealing the Queen of Hearts.

Next, I draw a card at random from pack B and shuffle it into pack C. Then, I turn up the top card of pack B, revealing another Queen of Hearts.

I now draw a random card from pack C and place it at the bottom of pack A.

What is the probability that the card at the top of pack C is the Ace of Spades?