## Lesson 11

In this lesson, we will cover more on canonical forms. First recall that for *m*↑ + (**n*) with *m* > 0, this game is positive except when (*m*, *n*) = (1, 1). Let’s consider the canonical forms of these games.

First, we know that ↑ = {0 | *}.

What about ↑* = ↑ + * ? By definition, we get:

since * + * = 0 < ↑. Now let’s consider the *Left* option ↑. This has a *Right* option *. Now is it true that * ≤ ↑*. Of course! So we can replace the *Left* option ↑ by all the *Left* options of *. This gives:

We’ll leave it to the reader to check that further simplification is impossible. So we have the canonical form of ↑*.

Next, what about 2↑ = ↑ + ↑? Again, by definition,

Let’s see if move reversal takes place. *Left*‘s option to ↑ has the *Right* option *. Now is it true that * ≤ 2↑ ? Yup! So we can replace *Left*‘s option ↑ by all *Left* options of *. This gives .

Now let’s consider the *Right* option to ↑*. Since we already saw that ↑* = {0, * | 0}, this has *Left* options 0 and *. Now is it true that 0 (or *) ≥ 2↑ ? Nope to both cases! So we obtain the following canonical form:

.

We’ll leave it to the reader to calculate the canonical forms of the following and find a pattern among them (see exercise 2).

- 2↑ + *
- 3↑
- 3↑ + *
- 4↑
- 4↑ + *

Next, let’s consider *G* = ↑ + *2. By definition, we have:

.

Since ↑ > *2 we can erase the *Left* option *2. Likewise, since *3 < ↑ and *3 < ↑* we can erase the *Right* options ↑ and ↑*. This leaves:

Any move reversals possible? Let’s consider the *Left* option ↑, which has *Right* option *. Since * < *G*, move reversal happens and we replace ↑ by the *Left* options of *, thus giving . By the same token, the *Left* option ↑* has *Right* option 0 < *G*, so we shall replace ↑* by the *Left* options of 0, i.e. nothing! This gives us:

.

Next, calculate the canonical forms of the following and find a pattern among them (see exercise 3).

- ↑ + *3
- ↑ + *4
- ↑ + *5

### More Domineering

It turns out for small *m* and *n*, the *m*-by-*n* Domineering board has rather nice canonical forms. The following can be calculated on a computer using cgsuite:

1 | 2 | 3 | 4 | 5 | 6 | |

1 | 0 | |||||

2 | 1 | ±1 | ||||

3 | 1 | {1/2 | -2} | ±1 | |||

4 | 2 | +_{2} |
3/2 | G |
||

5 | 2 | 1/2 | 1 | -1 | 0 | |

6 | 3 | {1 | -1+(+_{2})} |
{7/2 | 1} | #&!?* | 3/2 | Out of mem |

7 | 3 | {1/2 | -3/2} | {3 | 3/4} | -1 |

where

- –
_{2}= {{2 | 0} | 0}, - +
_{2}= – (-_{2}) = {0 | {0 | -2}}, *G*= {0,*H*| 0, –*H*}, and*H*= {{2|0}, 2+(+_{2}) | {2|0}, –_{2}}.

There are also many sequences of Domineering configurations which admit a general formula. E.g.:

### More on Toppling Dominoes

Recall the game of Toppling Dominoes in lesson 9. We already know the following games:

- Contiguous row of
*n**Light*dominoes :*n*. - Contiguous row of
*n**Light*dominoes, followed by*m*dominoes : {*n*-1 | -(*m*-1)}.

As a special case, a *Light* domino placed beside a *daRk* domino forms the game {0 | 0} = *. Now we can calculate the following games:

See if you can find a pattern for this game (see exercise 1).

### Exercises

- Guess the canonical forms of the following games (the first game has
*n*+1*Light*and*n**daRk*dominoes; the second game has*n**Light*and*n**daRk*dominoes). Can you prove it?

- Write down the canonical forms of
*n*↑ and*n*↑ + *. Prove it. [ F*or this and the next two problems, take note of the fact that ↑* is fuzzy.*] - Write down the canonical form of ↑ + *
*m*. Prove it. - Even more generally, write down the canonical form of
*n*↑ + **m*. Prove it. - Given the following two Domineering games:

simplify the following into canonical form:

- Given the following Toads-and-Frogs games:

simplify the following games into canonical forms: