Solving Puzzles Provably: Dominos and Towers
Koray Karabina, [email protected]
Domino Tiling: Let Rn be a 2 × n rectangular region divided into 1 × 1 squares. What
is the number of different domino tilings of Rn ? A domino is a rectangular 2 × 1 tile with a
line dividing its face into two square ends.
(a) R6
(b) Domino
(c) Domino
Figure 2: Domino tiling
Figure 3: A tiling of R6
(1) Let An be the number of different domino tilings of Rn . What is A6 ?
Koray Karabina
1
Gold Coast Math Teachers’ Circle, Florida Atlantic University, 2014
Problem solving: If a problem involves a number (parameter), then it is a useful
strategy to attack other (easier) variants of the problem with smaller parameters. Reformulate the problem.
(2) What is A1 ? What is A2 ? What is A3 ?
Rectangle parameters
R1 : 2 × 1
R2 : 2 × 2
R3 : 2 × 3
Number of tilings
A1 =
A2 =
A3 =
(3) Can you see any pattern among A1 , A2 , and A3 ? How do you explain it?
(4) Can you guess A4 ? Can you guess An ? Can you verify your guess for A4 , A5 ?
Conjecture: A4 =
and An =
Evidence:
Verification:
Koray Karabina
2
Gold Coast Math Teachers’ Circle, Florida Atlantic University, 2014
(5) What is A4 ? Can you see any pattern among A1 , A2 , A3 , A4 ? How do you explain it?
Conjecture: A4 =
and An =
Evidence:
Verification:
(6) Can you guess A5 ? Can you guess An ? State a theorem and prove it.
Theorem: A5 =
and An =
Proof:
(7) Fill in the following table:
A1
A2
A3
A4
A5
A6
A7
Does the sequence look
familiar?
Figure 4: Fibonacci numbers
Koray Karabina
3
Gold Coast Math Teachers’ Circle, Florida Atlantic University, 2014
Koray Karabina
4
Gold Coast Math Teachers’ Circle, Florida Atlantic University, 2014
The Tower of Hanoi: The Tower of Hanoi Puzzle consists of 3 posts in a line and n discs.
All the discs are of different size and all have a hole in the center so that the discs can be
placed over the posts. The discs are all stacked on one post, in order of size, with the largest
disc on the bottom and the smallest disc on the top. The object of the puzzle is to move the
stack of discs from one post to another post by moving one disc at a time so that no disc is
ever placed on top of a smaller disc.
Origins The puzzle was first publicized in the West by the French mathematician Édouard
Lucas in 1883. There is a story about an Indian temple in Kashi Vishwanath which contains
a large room with three time-worn posts in it surrounded by 64 golden disks. Brahmin
priests, acting out the command of an ancient prophecy, have been moving these disks, in
accordance with the immutable rules of the Brahma, since that time. The puzzle is therefore
also known as the Tower of Brahma puzzle. According to the legend, when the last move of
the puzzle will be completed, the world will end. It is not clear whether Lucas invented this
legend or was inspired by it. 1
(1) Is the world ever going to end?
Let Tn denote the Tower of Hanoi puzzle with n discs. Can you solve T64 ?
(2) When is the world going to end?
Assuming that Tn has a solution, let An denote the minimum number of “moves” to
solve Tn . What is A64 ?
Problem solving: If a problem involves a number (parameter), then it is a useful
strategy to attack other (easier) variants of the problem with smaller parameters.
1
http://en.wikipedia.org/wiki/Tower_of_Hanoi
Koray Karabina
5
Gold Coast Math Teachers’ Circle, Florida Atlantic University, 2014
(3) What is A1 ? What is A2 ? What is A3 ?
Tower parameters
T1 : 1 disc
T2 : 2 discs
T3 : 3 discs
Number of moves
A1 =
A2 =
A3 =
(4) Can you see any pattern among A1 , A2 , and A3 ? How do you explain it?
(5) Can you guess A4 ? Can you guess An ? Can you verify your guess for A4 ?
Conjecture: A4 =
and An =
Evidence:
Verification:
Koray Karabina
6
Gold Coast Math Teachers’ Circle, Florida Atlantic University, 2014
(6) Can you guess A7 , A64 , An ? Can you actually solve T7 , T64 , Tn ? Describe an algorithm
(method) that solves T7 (or even Tn ).
Algorithm:
(7) When is the world going to end? Assume that you start solving the puzzle T64 right
now, and you can move disks at a rate of one per second.
Koray Karabina
7
Gold Coast Math Teachers’ Circle, Florida Atlantic University, 2014
The Tower of Hanoi II Consider a variant of the Towers of Hanoi Puzzle, where we
change the rules slightly. In this case, we can only move a disc to an adjacent post (the end
posts are only adjacent to the middle post and not to each other). Suppose all the discs start
at one end. Show that they can all be moved to the post at the other end, and determine
the number of moves required.
Let Tn∗ denote the The Tower of Hanoi II with n discs, and let A∗n denote the minimum
number of “moves” to solve Tn∗ .
(1) Is solving Tn∗ easier than solving Tn ?
(2) What is A∗1 , A∗2 ?
Tower parameters
T1∗ : 1 disc
T2∗ : 2 discs
Number of moves
A∗1 =
A∗2 =
(3) Can you guess A∗3 ? Can you guess A∗n ? Can you verify your guess for A∗3 ?
Conjecture: A∗3 =
and A∗n =
Evidence:
Verification:
8
Koray Karabina
Gold Coast Math Teachers’ Circle, Florida Atlantic University, 2014
∗
(4) Can you guess A∗7 , A∗64 , A∗n ? Can you actually solve T7∗ , T64
, Tn∗ ? Describe an algorithm
(method) that solves T7∗ (or even Tn∗ ).
Algorithm:
Koray Karabina
9
Gold Coast Math Teachers’ Circle, Florida Atlantic University, 2014
Koray Karabina
10
Gold Coast Math Teachers’ Circle, Florida Atlantic University, 2014
Worksheet
(a) R6
(b) Dominos
(c) R6
(d) Dominos
(e) R6
(f) Dominos
(g) R6
(h) Dominos
Koray Karabina
11
Gold Coast Math Teachers’ Circle, Florida Atlantic University, 2014