Problems

How many \(7\)-digit numbers, larger than \(6\) million, are there such that the product of their digits is \(42\)?

Here is a half of a magic trick to impress your friends. Take the Ace to Seven of hearts and order them numerically in a pile with Ace at the bottom and Seven at the top. Do the same with the Ace to Seven of spades. Put the pile of spades on top of the pile of hearts. Flip the \(14\)-card deck so the cards are now face down.

Make a cut anywhere. Deal out \(7\) cards from the top into a pile, so that you now have two piles of equal size again. Let us refer to the motion of taking the top card of a pile and putting it to the bottom as making a swap. Making two swaps means doing the motion twice, NOT taking the top two cards at the same time.

Your friend chooses two nonnegative numbers that add up to \(6\), for example \(4\) and \(2\). Make \(4\) swaps on the first pile and \(2\) swaps on the second pile. Remove the top card of each pile, so that we now have two piles of six cards.

Your friend now chooses two nonnegative numbers that add up to five and repeat the same procedure. We will continue doing this, but at each turn, the total number of swaps decreases by one. Eventually, we should have one card remaining in each pile. It turns out to be a matching pair.

How does the trick work?

Many magicians can perform what is known as the Faro shuffle. Actually there are two kinds of Faro shuffle: the in Faro shuffle and the out Faro shuffle.

Let us assume that the deck has an even number of cards. The first step of the Faro shuffle is to divide the deck into two smaller decks of equal size. One deck consist of the top half of the original deck in their original order. The other deck consists of the bottom half of the original deck in their original order.

The second step of the Faro shuffle is to interweave the two decks, so that each card is above and beneath a card of the opposite deck. This is where the in Faro and the out Faro differ: the in Faro changes the top and bottom cards of the original deck (from before step one) while the out Faro retains the original top and bottom cards.

Show that only 8 out Faro shuffles are needed to return a standard 52 card deck back into its original position.

How can you move the top card to any position in an even size deck using only Faro shuffles?

There are six symmetries of an equilateral triangle: three reflections, and three rotations (thinking of the identity as one the rotations). Label the three reflections \(s_1\), \(s_2\) and \(s_3\). Label the identity by \(e\), rotation by \(120^{\circ}\) as \(r_1\), and rotation by \(240^{\circ}\) clockwise as \(r_2\).

Note the following definition: Each symmetry has an inverse. Suppose we apply symmetry \(x\). Then there is some symmetry we can apply after \(x\), which means that overall, we’ve applied the identity. What are the inverses of \(r_1\) and \(s_1\)?

Think about the symmetries of an equilateral triangle. Is applying rotation by \(120^{\circ}\), and then reflecting in the vertical median the same as applying these two symmetries the other way around?

Let \(n\ge3\) be a positive integer. A regular \(n\)-gon is a polygon with \(n\) sides where every side has the same length, and every angle is the same. For example, a regular \(3\)-gon is an equilateral triangle, and a regular \(4\)-gon is a square.

What symmetries does a regular \(n\)-gon have, and how many?

Show that there are infinitely many composite numbers \(n\) such that \(3^{n-1}-2^{n-1}\) is divisible by \(n\).

Show that there are infinitely many numbers \(n\) such that \(2^n+1\) is divisible by \(n\). Find all prime numbers, that satisfy this property.

If \(k>1\), show that \(k\) does not divide \(2^{k-1}+1\). Find all prime numbers \(p,q\) such that \(2^p+2^q\) is divisible by \(pq\).