Concrete Mathematics Chapter 1 Warmups

It took me far longer than it should have, and I had a very partial success; I guess my excuse is that my brain was still cold…

At least I can claim I did try to solve all the exercises; I really spent hours on this.

Warmups

Horse colour

I kind of botched this one, as I tried to answer it even before reading the chapter… and my first instinct was that such a use of induction (taking numbered subsets) was invalid.

Of course it is not. This is a perfectly valid approach, but, as the book states, in the present case it breaks down for $n=2$.

Properly expressed with math notation, it becomes clear that the “same colour” concept is a binary relation (a reflexive, symmetric and transitive one). The key is binary: if every pair of horses were the same colour, then induction could be used.

Tower of Hanoi Variation

The description in the book is somewhat confusing, as it states the restriction in terms of absolute positions (that is, no direct move between left peg and right peg), rather than relative (if you want to move a disc between peg $A$ and peg $B$, you must first move it to peg $C$, then to peg $B$).

The first approach does not work (that is, it is impossible to solve the problem under these conditions), but obviously the authors meant the second approach.

Number of moves

This variation can be solved using the exact same tools as the original problem.

Assuming we want to move a stack from $A$ to $B$, using $C$ as transfer peg: a single disc can be moved in $2$ steps ($A$ to $C$, $C$ to $B$); to move more than $1$, you first need to move the $n-1$ from $A$ to $B$, then move $1$ disk from $A$ to $C$, move the $n-1$ discs from $B$ back to $A$, move the one disc from $C$ to $B$, and finally move the $n-1$ discs from $A$ to $B$.

More concisely:

\begin{aligned} T_1 &= 2&&\text{base case}\\ T_n &= T_{n-1} + 1 + T_{n-1} + 1 + T_{n-1}\\ & = 3T_{n-1} + 2&&\text{recurrence equation} \end{aligned}

Using the exact same method as in the book, let’s define $T_n + 1= U_n$:

\begin{aligned} U_1 &= T_1 + 1\\ & = 3\\ U_n &= T_n + 1\\ & = 3(U_{n-1} -1) + 3\\ & = 3U_{n-1} - 3 + 3\\ & = 3U_{n-1} \end{aligned}

Then, $U_n = 3^n$, and $T_n = 3^n-1$.

Arrangements

As discs must be sorted, to describe an arrangement it is enough to list the peg for each disc. As there are $3$ pegs, this means there are $3^n$ different arrangements.

The variation takes $3^n-1$ moves, but counting the starting position as well, this means $3^n$ different positions, which is the same as the total number of arrangements.

Tower of Hanoi, Initial Setup Variation

Once again, by induction: to move a disk to peg $B$:

Base case: moving the smallest disc takes at most $1$ move ($0$ if it is already on peg $B$), so $T_1 \le 1$; Recurrence: to move the disc of size $n$, assuming it is on $A$, we need to move all the smaller discs to $C$ (to clear both $A$ and $B$), then move the disc of size $n$, and finally move all the smaller discs to $B$. Calling the clearing operation $Cl_n$, we have $T_n \le Cl_{n-1} + 1 + T_{n-1}$.

A moment of thought is enough to realise that $Cl_n$ amounts to the same operation as $T_n$ (that is, move each disc to a specific peg, no matter where it currently is), so we have $Cl_n = T_n$, and therefore $T_n \le 2T_{n-1} + 1$, which is the same recurrence equation as the original problem.

Therefore there is no position that is more that $2^n-1$ moves from the target position.

Venn Diagram with 4 circles

I completely failed to solve this one, even though I spent most of the time on this problem alone. I had the intuition that it could not be done; I also found that the maximum number of regions would be 14, but not matter what I tried, I could not prove it.

I tried to use Geometry, hoping that a minimal list of constraints on the circles would prove that some of the regions that should be restricted to two circles were in fact always covered by three or more.

Eventually, when I gave up and looked at the solution, I still could not understand it. So a circle can only intersect another one in at most 2 points. OK, so what?

After more research (the Google kind, this time), I found this paper which explains why. Each intersection point creates a single new region. Although once again I have no intuition I can trust in this domain, in this case the reasoning seems similar enough to intersecting lines that I feel somewhat confident.

So the above observation gives a recurrence equation:

\begin{aligned} C_1 &= 2\\ C_n &= C_{n-1} + 2(n-1) \end{aligned}

Already, we have that $C_4 = 14$, which is less than the required $16$ for a Venn diagram (and according to this document), four circles form a Euler diagram, not a Venn diagram.

Clearly a triangular number sequence is hiding in there. The recurrence equations above can be rewritten as

\begin{aligned} C_n &= 2+\sum_{i=1}^{n}2(i-1)\\ &= 2+2\sum_{i=0}^{n-1}i\\ &= 2+2\frac{n(n-1)}{2}\\ &= n^2-n+2 \end{aligned}

Bounded Regions in the Plane

Another one where my intuition for Geometry completely failed me. I had a correct start, identifying that each new line intersecting the existing ones at $k$ points could at best create $k-1$ new bounded regions, but when I try to check this I fumbled.

Yet the reason it simple: a line intersecting 2 others will either define a bounded triangle, or cut an existing bounded region in two.

The new bounded regions are not made of arbitrary triple of lines, but are next to each others in the plane; really this is similar to the fence problem. So a line cutting $k$ other lines will create at best $k-1$ new bounded region. The equality is achieved if there are no parallel lines, and all the intersection points are distinct.

As the book observes, each new line will also add two new unbounded regions (the original problem had that a new line would create $k+1$ new regions).

Once again, the triangular number sequence is not far:

\begin{aligned} B_i & = 0 &&\text{for 1 \le i \lt 3}\\ B_3 & = 1\\ B_n & = B_{n-1} + n - 2\\ & = \sum_{i=2}^{n} i-2\\ & = \sum_{i=0}^{n-2} i\\ & = \frac{(n-1)(n-2)}{2}\\ & = S_{n-2} \end{aligned}

Invalid Recurrence

The recurrence for $H$ has a number of problems. The one I found is that it only establishes the induction hypothesis for going from an even number to an odd one; nothing can be said for going from an odd number to an even one (and indeed, the hypothesis breaks then).

As the book mentions, another problem is the base case, which is incompatible with the induction hypothesis.

Wrapping up

I spent way too much time on these exercises, but most of it was on exercises with a geometric nature: I could not find an algebraic description of these problems that would be suitable for the kind of treatment this chapter is about. But once I had the equations, I was able to solve the problems without trouble.

Next, the homework exercises.