**2. ****Conway**** Squares and Their Representations**** **

The colored shapes in the table below will be used to represent sixteen objects, with each one having four visual attributes: color, border, shape, and size. As indicated in the table, each attribute has two values: color can be green or yellow, border can be fat or thin, shape can be square or triangle, and size can be small or large.

**Illustration 2.1 **

**Definition
2.1.** The collection of sixteen colored
shapes in Illustration 2.1 will be called *Conway** items**.* A set of four *Conway** property *just in case both values of each attribute (color, border,
shape, and size) occur exactly twice. Notice that both the left and right main
diagonals of the table in Illustration 2.1 each have the

A *Conway*** square**
is any rearrangement of the table above in which each
of the four rows, the four columns, and the two (left and right) main diagonals
have the

We note that our use of *perfect* just in case nim-adding any number from 0 to 15 to every
entry of it produces a magic square. Though only base 10 magic squares are
considered there, this notion of preservation under nim-addition gave us a hint
that considering a base 2 representation (see Definition 2.2 below) might be
interesting and fruitful. Furthermore, in Section 3 of this paper we show how
to use this base 2 representation to generate all

**Exercise
2.1.** Play the

**Exercise
2.2.** Play the Conway game starting with
both diagonals as in Exercise 2.1, except that this time interchange the first
two objects on the right main diagonal with the two objects below them on this
diagonal. So, the new right main diagonal starting from the upper right and
proceeding down and to the left should be: the large yellow square with thin
border, the small yellow square with fat border, the large green triangle with
thin border, and the small green triangle with fat border.

**Definition
2.2.** As with Euler squares, we also want
to numerically code

The tables below illustrate this correspondence
between

**Base 2 and Base 10 Coding of ****Conway**** Items**

**Illustration 2.2 **

If the Conway items in a Conway square are
numerically coded using the base 2 (base 10) coding, then the resultant
numerical square will be called **a base
2**

**Base 2 and Base 10 Coding of this ****Conway Square**** **

**Illustration 2.3 **

**Exercise
2.3.** Code each of the

**Theorem
2.3.** Every base 10

**Exercise
2.4.** Try to explain why Theorem 2.3 holds
by finding numerical properties of the base 2 coding of a

**Theorem
2.4.** Let f be the ** natural mapping** of Euler
items onto

(1) If E is an Euler square, then f [E] (the square obtained by applying f to each item in E) is a

(2) The embedding in (1) is not an onto mapping, i.e. every Euler square gets mapped to a

**Exercise
2.4.**

(a) Using the mapping in Theorem 2.4 (1), map all of the Euler squares in
Exercise 1.5 into

(b) After doing (a), try to explain why this mapping works by describing (for
example) how a row of Euler items satisfying the Euler property must get mapped
to a row of

(c) In addition, verify Theorem 2.4 (2) by explaining why the

Hint for (b) and (c): It is easier to try to use the numerical representations
of these squares, rather than directly using their visual representations in
terms of Euler and Conway items. If you get stuck here, there is more helpful
information below.

One interesting aspect of Theorem 2.4 is that we have been able to state it in terms of visual objects: Euler and Conway items, and in terms of visual magic squares: Euler and Conway squares. Though this visual approach is very appealing in that it allows us to easily create and recognize these visual magic squares, it is somewhat more difficult to use in actually proving results about these squares. Often that's the trade off one deals with in mathematics. Sometimes a visual representation is easier to use to see what's going on intuitively, but an appropriate numerical representation provides an immediate well-developed theoretical framework within which we can more easily state and verify important properties and results. Our intention in setting Exercise 2.4 was for the reader to experience this situation in doing math, and hence realize the necessity and usefulness of the numerical representations. The next theorem is essentially the numerical equivalent of Theorem 2.4.

**Theorem
2.5.** Let e be an Euler item, let [i, j] be
its base 4 representation, and let [k, l, m, n] be the base 2 representation of
[i, j]. Further, for each such e let g be the natural mapping sending [i, j] to
[k, l, m, n].

(1) If E is the base 4 representation of an Euler square, then g[E] (the square
obtained by applying g to each entry in E) is the base 2 representation of a

(2) The embedding in (1) is not an onto mapping, i.e. every base 4 Euler square
gets mapped to a base 2 Conway square, but some base 2 Conway square doesn't
come from a base 4 Euler square.

**Exercise
2.5.** Explain why Theorem 2.5 holds. Again,
note that Theorem 2.5 is just a restatement of Theorem 2.4 in terms of
numerical representations of Euler and Conway squares, and so this exercise is
really just a continuation of Exercise 2.4 (b) and (c) with a more detailed
hint on what numerical representations to use.

**Notation.** In the following development we will use 'Euler square'
and 'Conway square' to refer to both their corresponding visual item representations,
and to their various numerical representations in different bases. With this
notational convention we now see (can say) that it follows from Theorems 2.4
and 2.5 that every Euler square is a Conway square, but not vice versa. We
noted in Remark 1.7 that it follows from Theorem 1.6 that there are 144
essentially different Euler squares. We know that the

**Definition
2.6.** Two ** similar**
or

In Illustration 2.4
below, we have labeled each

**Illustration 2.4 **

Using the numerical labels in Illustration 2.4, and considering the left main diagonal, we see that item 0 differs from itself in 0 attributes, item 0 differs from item 5 in 2 attributes (border and size), item 0 differs from item 10 in 2 attributes (color and shape), and item 0 differs from object 15 in all 4 attributes (color, border, shape, and size). So, the items on the left main diagonal, i.e. items 0, 5, 10, and 15, are all in the same similarity class.

Furthermore, items 0 and 15 are complementary, since they differ in all 4 attributes, and items 5 and 10 are complementary as well. In addition, items 0 and 5 are bicomplementary, and so are items 0 and 10, items 15 and 5, and items 15 and 10.

**Exercise
2.6.** Verify that the similarity relation ~
defined on

**Exercise
2.7.** Verify that each similarity class can
be written in terms of anyone of its elements as follows: [a] = {a, a'} È a*,
i.e. the similarity class of a is comprised of a, the complement of a, and all
the bicomplements of a.

**Exercise
2.8.** Write a clear, detailed, math
argument explaining why complementary items have the same bicomplements, i.e.
if a' = b, then a* = b*.

Next, we have the same

**Illustration 2.5 **

**Definition
2.7.** **( Diagonal
Types)** We will now define three different diagonal types of

1. A *Type
D1*** ****Conway**** square**
is a

**Illustration 2.6 **

Note that for ease of viewing, we have partitioned the pattern for a Type D1 Conway square (the one on the left of the equality in Illustration 2.6) into two partial squares, each depicting the location of one of the two similarity classes. We use a plus sign to indicate schematically how the two patterns are joined together.

Also, the name 'D1' corresponds to constructing a

2. A *Type
D2*** ****Conway**** square**
is a

**Illustration 2.7 **

Note that for ease of viewing, we have partitioned the pattern for a Type D2 Conway square (the one on the left of the equality in Illustration 2.7) into two partial squares, each depicting the location of one of the two similarity classes. We use a plus sign to indicate schematically how the two patterns are joined together.

Also, the name 'D2' corresponds to constructing a

3. A *Type D3*** ****Conway**** square** is a

**Illustration 2.8 **

Note that for ease of viewing, we have partitioned the pattern for a Type D3 Conway square (the one on the left of the equality in Illustration 2.8) into two partial squares, each depicting the location of one of the two similarity classes. We use a plus sign to indicate schematically how the two patterns are joined together.

Also, the name 'D3' corresponds to constructing a

**Theorem
2.8.** There are 384

**Corollary
2.9.** Each Diagonal Type Conway square is
preserved under the eight symmetry transformations described in Remark 1.7,
i.e. the *symmetries of** the*

**Sketch of
Proof of Theorem 2.8: Type D1 ****Conway**** squares.**

Note that we will use a

So, we begin by placing an arbitrary

Finally, it can be seen by playing the

**Sketch of
Proof of Theorem 2.8: Type D2 ****Conway**** squares.**

We begin by placing an arbitrary

Finally, it can be seen by playing the

**Sketch of
Proof of Theorem 2.8: Type D3 ****Conway**** squares.**

We begin by placing an arbitrary

Finally, it can be seen by playing the

**Exercise
2.9.** If you haven't done so already, play
the Conway game while reading each proof sketch above, mirroring all the steps
in a specific case for each of the three Diagonal Types. As a result, you will
have constructed a

**Exercise
2.10.** By viewing Illustrations 2.6, 2.7,
and 2.8, convince yourself that each of the eight symmetries of the square do
indeed preserve each of the Diagonal Types. And so, as stated in Corollary 2.9,
ignoring these symmetries there really are only 48 = 384/8 essentially
different

**Remark
2.10.** Note that if we wished, we could
enumerate all the Diagonal Type Conway squares simply by playing the

**Definition
2.11. ( Row Types)** We will now
define three different row types of

1. A *Type
R1*** ****Conway**** square** is a

**Illustration 2.9 **

Note that for ease of viewing, we have partitioned the pattern for a Type R1 Conway square (the one on the left of the equality in Illustration 2.9) into two partial squares, each depicting the location of one of the two similarity classes. We use a plus sign to indicate schematically how the two patterns are joined together.

Also, the name 'R1' corresponds to constructing a

2. A *Type
R2*** ****Conway**** square** is a

**Illustration 2.10 **

Note that for ease of viewing, we have partitioned the pattern for a Type R2 Conway square (the one on the left of the equality in Illustration 2.10) into two partial squares, each depicting the location of one of the two similarity classes. We use a plus sign to indicate schematically how the two patterns are joined together.

Also, the name 'R2' corresponds to constructing a

3. A *Type
R3*** ****Conway**** square**
is a

**Illustration 2.11 **

Note that for ease of viewing, we have partitioned the pattern for a Type R3 Conway square (the one on the left of the equality in Illustration 2.11) into two partial squares, each depicting the location of one of the two similarity classes. We use a plus sign to indicate schematically how the two patterns are joined together.

Also, the name 'R3' corresponds to constructing a

**Theorem
2.12.** There are 384

**Corollary
2.13.** Each Row Type Conway square is
preserved under the four symmetry transformations of the rectangle, i.e. a
rotation by 180 or 360 degrees, or a reflection in the horizontal or vertical
axis through the center. Hence, there are 96 *essentially different*
** essentially different** Conway Squares of Row Type R3, yielding a total of 384 essentially
different Row Type Conway Squares.

**Sketch of
Proof of Theorem 2.12: Type R1 ****Conway**** squares.**

Note that we will again use a

So, we begin by placing an arbitrary

Finally, it can be seen by playing the

**Sketch of
Proof of Theorem 2.12: Type R2 ****Conway**** squares.**

We begin by placing an arbitrary

Finally, it can be seen by playing the

**Sketch of
Proof of Theorem 2.12: Type R3 ****Conway**** squares.**

We begin by placing an arbitrary

In contrast to the other 5 types, it can be seen
by playing the

Hence, there are (16)(6)(4)(2) = 768 Type R3 Conway squares. ||

**Exercise
2.11.** If you haven't done so already, play
the Conway game while reading each part of the proof sketch of Theorem 2.12
above, mirroring all the steps in a specific case for each of the three Row
Types. As a result, you will have constructed a

**Exercise
2.12.** By viewing Illustrations 2.9, 2.10,
and 2.11, convince yourself that each of the four symmetries of the rectangle
do indeed preserve each of the Row Types. And so, as stated in Corollary 2.13,
ignoring these symmetries there really are only 96 = 384/4 essentially
different

**Remark
2.14.** Note that if we wished, we could
enumerate all the Row Type Conway squares simply by playing the