![Page 1: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/1.jpg)
Using Latin Squares to Test Video Games
Karen Meagher
Department of Mathematics and StatisticsUniversity of Regina
Regina, Saskatchewan, Canada
![Page 2: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/2.jpg)
A Dream Job
You are a video game tester and you have to test a retroversion of PacMan.
![Page 3: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/3.jpg)
A Dream Job
You are a video game tester and you have to test a retroversion of PacMan.
![Page 4: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/4.jpg)
Why Test?
The original version of PacMan was released with an error.
If you get to level 256 this is what you see,
They only allocated 8 bits ofmemory for the level counter.
In binary the largest 8 bitnumber is
11111111
which is equal to
27+26+25+24+23+22+21+20 = 255.
![Page 5: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/5.jpg)
Why Test?
The original version of PacMan was released with an error.If you get to level 256 this is what you see,
They only allocated 8 bits ofmemory for the level counter.
In binary the largest 8 bitnumber is
11111111
which is equal to
27+26+25+24+23+22+21+20 = 255.
![Page 6: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/6.jpg)
Why Test?
The original version of PacMan was released with an error.If you get to level 256 this is what you see,
They only allocated 8 bits ofmemory for the level counter.
In binary the largest 8 bitnumber is
11111111
which is equal to
27+26+25+24+23+22+21+20 = 255.
![Page 7: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/7.jpg)
Why Test?
The original version of PacMan was released with an error.If you get to level 256 this is what you see,
They only allocated 8 bits ofmemory for the level counter.
In binary the largest 8 bitnumber is
11111111
which is equal to
27+26+25+24+23+22+21+20 = 255.
![Page 8: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/8.jpg)
Why Test?
The original version of PacMan was released with an error.If you get to level 256 this is what you see,
They only allocated 8 bits ofmemory for the level counter.
In binary the largest 8 bitnumber is
11111111
which is equal to
27+26+25+24+23+22+21+20 = 255.
![Page 9: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/9.jpg)
Why Test?
The original version of PacMan was released with an error.If you get to level 256 this is what you see,
They only allocated 8 bits ofmemory for the level counter.
In binary the largest 8 bitnumber is
11111111
which is equal to
27+26+25+24+23+22+21+20 = 255.
![Page 10: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/10.jpg)
What to test
There are several “parameters” in the game.
1. PacMan’s speed: Slow, Medium, Fast2. Ghosts’ image:
Normal Blue Flashing
3. Maze: α, β, γ4. Fruit on screen:
Banana Orange Cherries
You need to test all of these parameters, so you make a chart
![Page 11: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/11.jpg)
What to test
There are several “parameters” in the game.
1. PacMan’s speed: Slow, Medium, Fast2. Ghosts’ image:
Normal Blue Flashing
3. Maze: α, β, γ4. Fruit on screen:
Banana Orange Cherries
You need to test all of these parameters, so you make a chart
![Page 12: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/12.jpg)
What to test
There are several “parameters” in the game.
1. PacMan’s speed:
Slow, Medium, Fast2. Ghosts’ image:
Normal Blue Flashing
3. Maze: α, β, γ4. Fruit on screen:
Banana Orange Cherries
You need to test all of these parameters, so you make a chart
![Page 13: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/13.jpg)
What to test
There are several “parameters” in the game.
1. PacMan’s speed: Slow, Medium, Fast
2. Ghosts’ image:
Normal Blue Flashing
3. Maze: α, β, γ4. Fruit on screen:
Banana Orange Cherries
You need to test all of these parameters, so you make a chart
![Page 14: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/14.jpg)
What to test
There are several “parameters” in the game.
1. PacMan’s speed: Slow, Medium, Fast2. Ghosts’ image:
Normal Blue Flashing
3. Maze: α, β, γ4. Fruit on screen:
Banana Orange Cherries
You need to test all of these parameters, so you make a chart
![Page 15: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/15.jpg)
What to test
There are several “parameters” in the game.
1. PacMan’s speed: Slow, Medium, Fast2. Ghosts’ image:
Normal
Blue Flashing
3. Maze: α, β, γ4. Fruit on screen:
Banana Orange Cherries
You need to test all of these parameters, so you make a chart
![Page 16: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/16.jpg)
What to test
There are several “parameters” in the game.
1. PacMan’s speed: Slow, Medium, Fast2. Ghosts’ image:
Normal Blue
Flashing
3. Maze: α, β, γ4. Fruit on screen:
Banana Orange Cherries
You need to test all of these parameters, so you make a chart
![Page 17: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/17.jpg)
What to test
There are several “parameters” in the game.
1. PacMan’s speed: Slow, Medium, Fast2. Ghosts’ image:
Normal Blue Flashing
3. Maze: α, β, γ4. Fruit on screen:
Banana Orange Cherries
You need to test all of these parameters, so you make a chart
![Page 18: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/18.jpg)
What to test
There are several “parameters” in the game.
1. PacMan’s speed: Slow, Medium, Fast2. Ghosts’ image:
Normal Blue Flashing
3. Maze:
α, β, γ4. Fruit on screen:
Banana Orange Cherries
You need to test all of these parameters, so you make a chart
![Page 19: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/19.jpg)
What to test
There are several “parameters” in the game.
1. PacMan’s speed: Slow, Medium, Fast2. Ghosts’ image:
Normal Blue Flashing
3. Maze: α, β, γ
4. Fruit on screen:
Banana Orange Cherries
You need to test all of these parameters, so you make a chart
![Page 20: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/20.jpg)
What to test
There are several “parameters” in the game.
1. PacMan’s speed: Slow, Medium, Fast2. Ghosts’ image:
Normal Blue Flashing
3. Maze: α, β, γ4. Fruit on screen:
Banana Orange Cherries
You need to test all of these parameters, so you make a chart
![Page 21: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/21.jpg)
What to test
There are several “parameters” in the game.
1. PacMan’s speed: Slow, Medium, Fast2. Ghosts’ image:
Normal Blue Flashing
3. Maze: α, β, γ4. Fruit on screen:
Banana
Orange Cherries
You need to test all of these parameters, so you make a chart
![Page 22: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/22.jpg)
What to test
There are several “parameters” in the game.
1. PacMan’s speed: Slow, Medium, Fast2. Ghosts’ image:
Normal Blue Flashing
3. Maze: α, β, γ4. Fruit on screen:
Banana Orange
Cherries
You need to test all of these parameters, so you make a chart
![Page 23: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/23.jpg)
What to test
There are several “parameters” in the game.
1. PacMan’s speed: Slow, Medium, Fast2. Ghosts’ image:
Normal Blue Flashing
3. Maze: α, β, γ4. Fruit on screen:
Banana Orange Cherries
You need to test all of these parameters, so you make a chart
![Page 24: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/24.jpg)
What to test
There are several “parameters” in the game.
1. PacMan’s speed: Slow, Medium, Fast2. Ghosts’ image:
Normal Blue Flashing
3. Maze: α, β, γ4. Fruit on screen:
Banana Orange Cherries
You need to test all of these parameters, so you make a chart
![Page 25: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/25.jpg)
Your Chart
Speed Ghosts Maze Type Fruit
test 1 slow α
test 2 slow α
test 3 slow α
test 4 slow β
test 5 slow β
test 6 slow β...
......
......
There are 3↑
speed
× 3↑
ghost
× 3↑
maze
× 3↑
fruit
= 81 different combinations!
Playing PacMan will become really boring before you finish.
![Page 26: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/26.jpg)
Your Chart
Speed Ghosts Maze Type Fruit
test 1 slow α
test 2 slow α
test 3 slow α
test 4 slow β
test 5 slow β
test 6 slow β...
......
......
There are 3↑
speed
× 3↑
ghost
× 3↑
maze
× 3↑
fruit
= 81 different combinations!
Playing PacMan will become really boring before you finish.
![Page 27: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/27.jpg)
Your Chart
Speed Ghosts Maze Type Fruit
test 1 slow α
test 2 slow α
test 3 slow α
test 4 slow β
test 5 slow β
test 6 slow β...
......
......
There are 3↑
speed
× 3↑
ghost
× 3↑
maze
× 3↑
fruit
= 81 different combinations!
Playing PacMan will become really boring before you finish.
![Page 28: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/28.jpg)
Your Chart
Speed Ghosts Maze Type Fruit
test 1 slow α
test 2 slow α
test 3 slow α
test 4 slow β
test 5 slow β
test 6 slow β
......
......
...
There are 3↑
speed
× 3↑
ghost
× 3↑
maze
× 3↑
fruit
= 81 different combinations!
Playing PacMan will become really boring before you finish.
![Page 29: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/29.jpg)
Your Chart
Speed Ghosts Maze Type Fruit
test 1 slow α
test 2 slow α
test 3 slow α
test 4 slow β
test 5 slow β
test 6 slow β...
......
......
There are 3↑
speed
× 3↑
ghost
× 3↑
maze
× 3↑
fruit
= 81 different combinations!
Playing PacMan will become really boring before you finish.
![Page 30: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/30.jpg)
Your Chart
Speed Ghosts Maze Type Fruit
test 1 slow α
test 2 slow α
test 3 slow α
test 4 slow β
test 5 slow β
test 6 slow β...
......
......
There are 3↑
speed
× 3↑
ghost
× 3↑
maze
× 3↑
fruit
= 81 different combinations!
Playing PacMan will become really boring before you finish.
![Page 31: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/31.jpg)
Your Chart
Speed Ghosts Maze Type Fruit
test 1 slow α
test 2 slow α
test 3 slow α
test 4 slow β
test 5 slow β
test 6 slow β...
......
......
There are 3↑
speed
× 3↑
ghost
× 3↑
maze
× 3↑
fruit
= 81 different combinations!
Playing PacMan will become really boring before you finish.
![Page 32: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/32.jpg)
Karen’s Solution to get to an Early Lunch
Speed Ghosts Maze Fruit
test 1: slow α
test 2: slow β
test 3: slow γ
test 4: medium β
test 5: medium γ
test 6: medium α
test 7: fast γ
test 8: fast α
test 9: fast β
This will test all nine possible pairs for any two parameters.
![Page 33: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/33.jpg)
Karen’s Solution to get to an Early Lunch
Speed Ghosts Maze Fruit
test 1: slow α
test 2: slow β
test 3: slow γ
test 4: medium β
test 5: medium γ
test 6: medium α
test 7: fast γ
test 8: fast α
test 9: fast β
This will test all nine possible pairs for any two parameters.
![Page 34: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/34.jpg)
Karen’s Solution to get to an Early Lunch
Speed Ghosts Maze Fruit
→ test 1: slow α
test 2: slow β
test 3: slow γ
test 4: medium β
test 5: medium γ
test 6: medium α
test 7: fast γ
test 8: fast α
test 9: fast β
This will test all nine possible pairs for any two parameters.
![Page 35: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/35.jpg)
Karen’s Solution to get to an Early Lunch
Speed Ghosts Maze Fruit
test 1: slow α
test 2: slow β
test 3: slow γ
→ test 4: medium β
test 5: medium γ
test 6: medium α
test 7: fast γ
test 8: fast α
test 9: fast β
This will test all nine possible pairs for any two parameters.
![Page 36: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/36.jpg)
Karen’s Solution to get to an Early Lunch
Speed Ghosts Maze Fruit
test 1: slow α
test 2: slow β
test 3: slow γ
test 4: medium β
test 5: medium γ
test 6: medium α
→ test 7: fast γ
test 8: fast α
test 9: fast β
This will test all nine possible pairs for any two parameters.
![Page 37: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/37.jpg)
Karen’s Solution to get to an Early Lunch
Speed Ghosts Maze Fruit
test 1: slow α
test 2: slow β
test 3: slow γ
test 4: medium β
→ test 5: medium γ
test 6: medium α
test 7: fast γ
test 8: fast α
test 9: fast β
This will test all nine possible pairs for any two parameters.
![Page 38: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/38.jpg)
Karen’s Solution to get to an Early Lunch
Speed Ghosts Maze Fruit
test 1: slow α
test 2: slow β
test 3: slow γ
test 4: medium β
test 5: medium γ
test 6: medium α
test 7: fast γ
→ test 8: fast α
test 9: fast β
This will test all nine possible pairs for any two parameters.
![Page 39: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/39.jpg)
Karen’s Solution to get to an Early Lunch
Speed Ghosts Maze Fruit
test 1: slow α
→ test 2: slow β
test 3: slow γ
test 4: medium β
test 5: medium γ
test 6: medium α
test 7: fast γ
test 8: fast α
test 9: fast β
This will test all nine possible pairs for any two parameters.
![Page 40: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/40.jpg)
Karen’s Solution to get to an Early Lunch
Speed Ghosts Maze Fruit
test 1: slow α
test 2: slow β
test 3: slow γ
test 4: medium β
test 5: medium γ
test 6: medium α
test 7: fast γ
test 8: fast α
→ test 9: fast β
This will test all nine possible pairs for any two parameters.
![Page 41: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/41.jpg)
Karen’s Solution to get to an Early Lunch
Speed Ghosts Maze Fruit
test 1: slow α
test 2: slow β
→ test 3: slow γ
test 4: medium β
test 5: medium γ
test 6: medium α
test 7: fast γ
test 8: fast α
test 9: fast β
This will test all nine possible pairs for any two parameters.
![Page 42: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/42.jpg)
Karen’s Solution to get to an Early Lunch
Speed Ghosts Maze Fruit
test 1: slow α
test 2: slow β
test 3: slow γ
test 4: medium β
test 5: medium γ
→ test 6: medium α
test 7: fast γ
test 8: fast α
test 9: fast β
This will test all nine possible pairs for any two parameters.
![Page 43: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/43.jpg)
Latin Squares
The first three columns can be written as a square:
Slow α β γMedium β γ αFast γ α β
Each symbol occurs once in every row and column.This is called a Latin square.
The fourth column can also be written as a square:
Slow
Medium
Fast
![Page 44: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/44.jpg)
Latin Squares
The first three columns can be written as a square:
Slow α β γMedium β γ αFast γ α β
Each symbol occurs once in every row and column.This is called a Latin square.
The fourth column can also be written as a square:
Slow
Medium
Fast
![Page 45: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/45.jpg)
Latin Squares
The first three columns can be written as a square:
Slow α β γMedium β γ αFast γ α β
Each symbol occurs once in every row and column.
This is called a Latin square.
The fourth column can also be written as a square:
Slow
Medium
Fast
![Page 46: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/46.jpg)
Latin Squares
The first three columns can be written as a square:
Slow α β γMedium β γ αFast γ α β
Each symbol occurs once in every row and column.This is called a Latin square.
The fourth column can also be written as a square:
Slow
Medium
Fast
![Page 47: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/47.jpg)
Latin Squares
The first three columns can be written as a square:
Slow α β γMedium β γ αFast γ α β
Each symbol occurs once in every row and column.This is called a Latin square.
The fourth column can also be written as a square:
Slow
Medium
Fast
![Page 48: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/48.jpg)
Latin Squares
The first three columns can be written as a square:
Slow α β γMedium β γ αFast γ α β
Each symbol occurs once in every row and column.This is called a Latin square.
The fourth column can also be written as a square:
Slow
Medium
Fast
![Page 49: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/49.jpg)
Greco-Latin Squares
We can put these to Latin squares together to make aGreco-Latin square.
Slow α, β, γ,
Medium β, γ, α,
Fast γ, α, β,
I get every maze/fruit combintation exactly once!
![Page 50: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/50.jpg)
Greco-Latin Squares
We can put these to Latin squares together to make aGreco-Latin square.
Slow α, β, γ,
Medium β, γ, α,
Fast γ, α, β,
I get every maze/fruit combintation exactly once!
![Page 51: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/51.jpg)
Greco-Latin Squares
We can put these to Latin squares together to make aGreco-Latin square.
Slow α, β, γ,
Medium β, γ, α,
Fast γ, α, β,
I get every maze/fruit combintation exactly once!
![Page 52: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/52.jpg)
Everyone loves Latin Squares!
Sudoko games are 9× 9 Latin squares.
![Page 53: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/53.jpg)
History
This was a “recreatational” math problem from 1725:
Take all aces, kings, queens and jacks from astandard deck of cards, and arrange them in a 4x4grid such that each row and each column contained allfour suits as well as one of each face value.
![Page 54: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/54.jpg)
History
This was a “recreatational” math problem from 1725:
Take all aces, kings, queens and jacks from astandard deck of cards,
and arrange them in a 4x4grid such that each row and each column contained allfour suits as well as one of each face value.
![Page 55: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/55.jpg)
History
This was a “recreatational” math problem from 1725:
Take all aces, kings, queens and jacks from astandard deck of cards, and arrange them in a 4x4grid
such that each row and each column contained allfour suits as well as one of each face value.
![Page 56: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/56.jpg)
History
This was a “recreatational” math problem from 1725:
Take all aces, kings, queens and jacks from astandard deck of cards, and arrange them in a 4x4grid such that each row and each column contained allfour suits as well as one of each face value.
![Page 57: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/57.jpg)
History
This was a “recreatational” math problem from 1725:
Take all aces, kings, queens and jacks from astandard deck of cards, and arrange them in a 4x4grid such that each row and each column contained allfour suits as well as one of each face value.
![Page 58: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/58.jpg)
Latin Squares are good for “art”
This is a 7× 7 Greco-Latin square (it is a baby blanket I madefor my Ph.D. supervisor’s baby).
![Page 59: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/59.jpg)
Latin Squares are good for “art”
This is a 7× 7 Greco-Latin square
(it is a baby blanket I madefor my Ph.D. supervisor’s baby).
![Page 60: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/60.jpg)
Latin Squares are good for “art”
This is a 7× 7 Greco-Latin square (it is a baby blanket I madefor my Ph.D. supervisor’s baby).
![Page 61: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/61.jpg)
36 Officers Problem
(from Wikipedia) The thirty-six officers problem is amathematical puzzle proposed by Leonhard Euler in 1782.
The problem asks if it is possible to arrange sixregiments consisting of six officers each of differentranks in a 6× 6 square so that no rank or regiment willbe repeated in any row or column.
This problem is asking for a6× 6 Latin square
In 1901 it was shown that it isnot possible to make a 6× 6Greco-Latin square!
![Page 62: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/62.jpg)
36 Officers Problem
(from Wikipedia) The thirty-six officers problem is amathematical puzzle proposed by Leonhard Euler in 1782.
The problem asks if it is possible to arrange sixregiments consisting of six officers each of differentranks in a 6× 6 square so that no rank or regiment willbe repeated in any row or column.
This problem is asking for a6× 6 Latin square
In 1901 it was shown that it isnot possible to make a 6× 6Greco-Latin square!
![Page 63: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/63.jpg)
36 Officers Problem
(from Wikipedia) The thirty-six officers problem is amathematical puzzle proposed by Leonhard Euler in 1782.
The problem asks if it is possible to arrange sixregiments consisting of six officers each of differentranks in a 6× 6 square so that no rank or regiment willbe repeated in any row or column.
This problem is asking for a6× 6 Latin square
In 1901 it was shown that it isnot possible to make a 6× 6Greco-Latin square!
![Page 64: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/64.jpg)
36 Officers Problem
(from Wikipedia) The thirty-six officers problem is amathematical puzzle proposed by Leonhard Euler in 1782.
The problem asks if it is possible to arrange sixregiments consisting of six officers each of differentranks in a 6× 6 square so that no rank or regiment willbe repeated in any row or column.
This problem is asking for a6× 6 Latin square
In 1901 it was shown that it isnot possible to make a 6× 6Greco-Latin square!
![Page 65: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/65.jpg)
36 Officers Problem
(from Wikipedia) The thirty-six officers problem is amathematical puzzle proposed by Leonhard Euler in 1782.
The problem asks if it is possible to arrange sixregiments consisting of six officers each of differentranks in a 6× 6 square so that no rank or regiment willbe repeated in any row or column.
This problem is asking for a6× 6 Latin square
In 1901 it was shown that it isnot possible to make a 6× 6Greco-Latin square!
![Page 66: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/66.jpg)
36 Officers Problem
(from Wikipedia) The thirty-six officers problem is amathematical puzzle proposed by Leonhard Euler in 1782.
The problem asks if it is possible to arrange sixregiments consisting of six officers each of differentranks in a 6× 6 square so that no rank or regiment willbe repeated in any row or column.
This problem is asking for a6× 6 Latin square
In 1901 it was shown that it isnot possible to make a 6× 6Greco-Latin square!
![Page 67: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/67.jpg)
10× 10 Latin Square
For a long time it was unknown if it was possible to build a10× 10 Greco-Latin square.
![Page 68: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/68.jpg)
10× 10 Latin Square
For a long time it was unknown if it was possible to build a10× 10 Greco-Latin square.
![Page 69: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/69.jpg)
10× 10 Latin Square
For a long time it was unknown if it was possible to build a10× 10 Greco-Latin square.
![Page 70: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/70.jpg)
A computer in 1959?
This is the type of computerthat was used to find the first10× 10 Greco-Latin square.
![Page 71: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/71.jpg)
A computer in 1959?
This is the type of computerthat was used to find the first10× 10 Greco-Latin square.
![Page 72: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/72.jpg)
Open Problems with Latin Squares
We look for ways to build them:
I can build an n × n Latin square for any n:
1 2 3 . . . n − 1 n2 3 4 . . . n 13 4 5 . . . 1 2...n 1 2 . . . n − 2 n − 1
I can also build an n × n Greco-Latin square if n is the power ofa prime number. It is easy, if you have a finite field!
![Page 73: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/73.jpg)
Open Problems with Latin Squares
We look for ways to build them:
I can build an n × n Latin square for any n:
1 2 3 . . . n − 1 n2 3 4 . . . n 13 4 5 . . . 1 2...n 1 2 . . . n − 2 n − 1
I can also build an n × n Greco-Latin square if n is the power ofa prime number. It is easy, if you have a finite field!
![Page 74: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/74.jpg)
Open Problems with Latin Squares
We look for ways to build them:
I can build an n × n Latin square for any n:
1 2 3 . . . n − 1 n2 3 4 . . . n 13 4 5 . . . 1 2...n 1 2 . . . n − 2 n − 1
I can also build an n × n Greco-Latin square if n is the power ofa prime number. It is easy, if you have a finite field!
![Page 75: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/75.jpg)
Open Problems with Latin Squares
We look for ways to build them:
I can build an n × n Latin square for any n:
1 2 3 . . . n − 1 n
2 3 4 . . . n 13 4 5 . . . 1 2...n 1 2 . . . n − 2 n − 1
I can also build an n × n Greco-Latin square if n is the power ofa prime number. It is easy, if you have a finite field!
![Page 76: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/76.jpg)
Open Problems with Latin Squares
We look for ways to build them:
I can build an n × n Latin square for any n:
1 2 3 . . . n − 1 n2 3 4 . . . n 1
3 4 5 . . . 1 2...n 1 2 . . . n − 2 n − 1
I can also build an n × n Greco-Latin square if n is the power ofa prime number. It is easy, if you have a finite field!
![Page 77: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/77.jpg)
Open Problems with Latin Squares
We look for ways to build them:
I can build an n × n Latin square for any n:
1 2 3 . . . n − 1 n2 3 4 . . . n 13 4 5 . . . 1 2
...n 1 2 . . . n − 2 n − 1
I can also build an n × n Greco-Latin square if n is the power ofa prime number. It is easy, if you have a finite field!
![Page 78: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/78.jpg)
Open Problems with Latin Squares
We look for ways to build them:
I can build an n × n Latin square for any n:
1 2 3 . . . n − 1 n2 3 4 . . . n 13 4 5 . . . 1 2...n 1 2 . . . n − 2 n − 1
I can also build an n × n Greco-Latin square if n is the power ofa prime number. It is easy, if you have a finite field!
![Page 79: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/79.jpg)
Open Problems with Latin Squares
We look for ways to build them:
I can build an n × n Latin square for any n:
1 2 3 . . . n − 1 n2 3 4 . . . n 13 4 5 . . . 1 2...n 1 2 . . . n − 2 n − 1
I can also build an n × n Greco-Latin square if n is the power ofa prime number.
It is easy, if you have a finite field!
![Page 80: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/80.jpg)
Open Problems with Latin Squares
We look for ways to build them:
I can build an n × n Latin square for any n:
1 2 3 . . . n − 1 n2 3 4 . . . n 13 4 5 . . . 1 2...n 1 2 . . . n − 2 n − 1
I can also build an n × n Greco-Latin square if n is the power ofa prime number. It is easy, if you have a finite field!
![Page 81: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/81.jpg)
Open Problems with Latin Squares
We try to count Latin squares:
There are two or order 2:
1 22 1
2 11 2
Order Number3 124 5765 1612806 812 851 200
![Page 82: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/82.jpg)
Open Problems with Latin Squares
We try to count Latin squares:
There are two or order 2:
1 22 1
2 11 2
Order Number3 124 5765 1612806 812 851 200
![Page 83: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/83.jpg)
Open Problems with Latin Squares
We try to count Latin squares:
There are two or order 2:
1 22 1
2 11 2
Order Number3 124 5765 1612806 812 851 200
![Page 84: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/84.jpg)
Open Problems with Latin Squares
We try to count Latin squares:
There are two or order 2:
1 22 1
2 11 2
Order Number3 124 5765 1612806 812 851 200
![Page 85: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/85.jpg)
Open Problems with Latin Squares
We classify Latin squares:
1 2 3 42 3 4 13 4 1 24 1 2 3
1 4 2 33 2 4 14 1 3 22 3 1 4
no transversal transversal
![Page 86: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/86.jpg)
Open Problems with Latin Squares
We classify Latin squares:
1 2 3 42 3 4 13 4 1 24 1 2 3
1 4 2 33 2 4 14 1 3 22 3 1 4
no transversal transversal
![Page 87: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/87.jpg)
Open Problems with Latin Squares
We classify Latin squares:
1 2 3 42 3 4 13 4 1 24 1 2 3
1 4 2 33 2 4 14 1 3 22 3 1 4
no transversal transversal
![Page 88: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/88.jpg)
Open Problems with Latin Squares
We classify Latin squares:
1 2 3 42 3 4 13 4 1 24 1 2 3
1 4 2 33 2 4 14 1 3 22 3 1 4
no transversal transversal
![Page 89: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/89.jpg)
Open Problems with Latin Squares
We classify Latin squares:
1 2 3 42 3 4 13 4 1 24 1 2 3
1 4 2 33 2 4 14 1 3 22 3 1 4
no transversal transversal
![Page 90: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/90.jpg)
Last Slide
Still LOTS of open problems with Latin squares andGraeco-Latin squares and experimental design.
I Build better test sets:I test sets that avoid combinations;I tests sets that balance other aspects;I tests sets that test only give pairs of the parameters.
I Latin Squares are important in other aspects of math:I finite fields—these are important for cyptography;I decompositions of graphs;I extremal combinatorics.
![Page 91: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/91.jpg)
Last Slide
Still LOTS of open problems with Latin squares andGraeco-Latin squares and experimental design.
I Build better test sets:
I test sets that avoid combinations;I tests sets that balance other aspects;I tests sets that test only give pairs of the parameters.
I Latin Squares are important in other aspects of math:I finite fields—these are important for cyptography;I decompositions of graphs;I extremal combinatorics.
![Page 92: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/92.jpg)
Last Slide
Still LOTS of open problems with Latin squares andGraeco-Latin squares and experimental design.
I Build better test sets:I test sets that avoid combinations;I tests sets that balance other aspects;I tests sets that test only give pairs of the parameters.
I Latin Squares are important in other aspects of math:I finite fields—these are important for cyptography;I decompositions of graphs;I extremal combinatorics.
![Page 93: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/93.jpg)
Last Slide
Still LOTS of open problems with Latin squares andGraeco-Latin squares and experimental design.
I Build better test sets:I test sets that avoid combinations;I tests sets that balance other aspects;I tests sets that test only give pairs of the parameters.
I Latin Squares are important in other aspects of math:
I finite fields—these are important for cyptography;I decompositions of graphs;I extremal combinatorics.
![Page 94: Using Latin Squares to Test Video Gamesuregina.ca/~meagherk/talk.pdf · Using Latin Squares to Test Video Games Karen Meagher Department of Mathematics and Statistics University of](https://reader033.vdocuments.us/reader033/viewer/2022051311/6041146babb81369c52528dd/html5/thumbnails/94.jpg)
Last Slide
Still LOTS of open problems with Latin squares andGraeco-Latin squares and experimental design.
I Build better test sets:I test sets that avoid combinations;I tests sets that balance other aspects;I tests sets that test only give pairs of the parameters.
I Latin Squares are important in other aspects of math:I finite fields—these are important for cyptography;I decompositions of graphs;I extremal combinatorics.