The Towers of Riga

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The Towers of Riga
Chiu Chang Mathematics Education
Foundation President Wen-Hsien SUN
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• There are twelve ancient towers in Riga. The base
  of each consists of three regular hexagons each
  sharing one side with each of the other two. On
  these regular hexagons are right prisms of
  varying heights, as indicated by the numbers in
  Figure 1 below.

3
Figure 1
D
C
A
B
G
F
H
E
I
J
L
K
4

• A legend says that the gods used magic to gather
  these twelve towers to the center of the town,
  and stored them in a large hexagonal box with a
  flat top and bottom and no spaces in between.

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Naming the Pieces

• The pieces are named by, starting at the shortest
  section, naming the height of each section in
  counterclockwise order. For example, the piece on
  the left would be referred to as (2, 5, 6).

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Naming the Pieces

• If there are two identical sections that are
  shortest, they are not separated, for example the
  bottom right piece would be referred to as (1, 1,
  2)

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• It is decided that the twelve towers are to be
  combined into a modern museum of height 7, with
  the pattern of the base as shown in either
  diagram of Figure 2. The pink central hexagon
  represents the elevator shaft.

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Figure 2
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• Clearly, six of the towers will have to be upside
  down, so that their flat bases form part of the
  flat top of the museum. Assuming that the
  practical difficulties can be overcome, the
  natural question is whether the towers actually
  fit together. In other words, is there a
  mathematical solution?

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• If both the top and the bottom of the apartment
  block follow the same one of the two patterns in
  Figure 2, then the twelve towers must form six
  complementary pairs, such as the (1,1,1) and the
  (6,6,6).

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• However, we can see quickly that this is not the
  case. For instance, we have a (4,6,6) but no
  (1,1,3).

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D
C
A
B
G
F
H
E
I
J
L
K
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• Thus we may assume that the base of the apartment
  block follows the pattern on the left of Figure
  2, and the top follows that on the right.

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• This implies that the twelve towers form six
  overlapping pairs, one right side up and the
  other upside down. Two of the hexagonal prisms of
  the latter are resting on two of those of the
  former, so that the two vertical neighbors have a
  combined height of 7.

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2. AL
1. AJ
A
1.
2.




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1. BG
2. BH
B
1.
2.




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1. BG
2. BH
B
3. BI
4. BJ
3.
4.




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1. BG
2. BH
B
3. BI
5. BL
4. BJ
5.


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1. CG
2. CJ
C
1.
2.




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1. CG
2. CJ
C
3. CI
3.(a)
3.(b)




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D
1. DI
2. DG
1.
2.(a)




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D
2. DG
1. DI
2.(b)
2.(c)




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E
EJ


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F
FK

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A B C D E F


G
J
K
G
G
J
I
I
J
I
L
L
J
H
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A B C D E F


G
J
K
L
G
G
I
L
I
I
H
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A B C D E F


J
K
L
H
G
G
I
I
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AL
(1, 1, 1)(6, 6, 6)(6, 1)

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BH
(1, 1, 2)(3, 5, 6)(3, 1)


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EJ
(1, 3, 4)(4, 6, 6)(6, 4)


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FK
(2, 2, 2)(5, 5, 5)(5, 2)

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Now we get the following accessories
(3, 1)
(5, 2)
(6, 4)
(6, 1)
D
G
I
C
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• We identify (2,2,2)(5,5,5),
• (1,3,4)(4,6,6),
• (1,1,2)(3,5,6) and
• (1,1,1)(6,6,6)
• with the dominoes (2,5), (4,6), (1,3) and (1,6),
  respectively.

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• Each of the dominoes (4,6) and (1,3) links a
  (1,6) with a (1,6).

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• In order to get the (2,5) into the domino ring,
  we need at least one domino in which one of the
  numbers is 2 or 5, and the other is 1 or 6. With
  this in mind, we now examine the remaining four
  towers. There are two possible couplings.

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C D


(i) CI?DG (ii) CG?DI
G
G
I
I
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(i)
DG
CI


False
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(ii)
DI
CG


False
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• Each of the dominoes (4,6) and (1,6) links a
  (1,3) with a (3,4). Method 1

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• Each of the dominoes (4,6) and (1,6) links a
  (1,3) with a (3,4). Method 2

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• In order to get the (2,5) into the domino ring,
  we need at least one domino in which one of the
  numbers is 2 or 5, and the other is 3 or 4. With
  this in mind, we now examine the remaining four
  towers. There are two possible couplings.

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C D


(i) CI?DG (ii) CG?DI
G
G
I
I
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(i)
DG
CI


False
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(ii)
DI
CG


OK!
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Method 1


7
7
7
7
7
7
7
7
7
7


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Method 1
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Method 1
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Method 2


7
7


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Method 2
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Method 2
6
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J
J
C
C
L
L
K
K
H
H
D
D
E
E
A
A
G
G
B
B
F
F
I
I
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Remark
It should be mentioned that uniqueness is up
to symmetry. Had we assumed that the base of the
apartment block follows the pattern on the right
of Figure 2, and the top follows that on the
left, we would obtain the solution in the right
of last page. We do not consider these two
solutions distinct.
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All Combination not higher than 6
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History
The idea for the Tower of Riga was found in a
32-page pamphlet Jug of Diamonds (in Russian),
published by the Ukrainian puzzlist Serhiy
Grabarchuk in Uzhgorod, 1991. The towers there
are all non-symmetric, but it has many solutions.
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History
The late Latvian mathematician and computing
scientist Alberts Vanags found by hand a set of
towers with only three solutions. Later, Atis
Blumbergs, a retired Latvian physician, found by
computer a set with a unique solution. This feat
was duplicated by Marija Babica, a master degree
student at the University of Latvia. However,
these sets contain many symmetric towers.
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History
Our set, found by Andris Cibulis, has only six
symmetric towers. It was used as his Exchange
Gift at the 25th International Puzzle Party in
Helsinki in July, 2005. Afterwards, the right to
manufacture it is granted to Chiu Chang
Mathematics Books and Puzzles in Taipei, a
company affiliated with Chiu Chang Mathematics
Foundation.
62
History
A power-point demonstration of the solution may
be found on the Foundation's website http//www.
chiuchang.org.tw/download/catalog/rigatower.ppt
63
History
In 2006, Marija Babica found the following set of
towers containing only four symmetric ones and
yet has a unique solution (1,1,1), (3,3,3),
(4,4,4), (6,6,6), (1,2,3), (1,3,5), (1,5,6),
(1,6,4), (2,3,5), (2,5,4), (2,6,4) and (3,4,6).
Using our approach, the reader should have little
difficulty solving this version of the Towers of
Riga.