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Math behind the Tiles



Why do 7 numbers create 28 tiles?


After all, with six numbers, we get 36 combinations with two dice.



Let's start with the simplest situation.  If we have one number, we get one tile.


One number, 1 tile

0,0



Two numbers, 3 tiles

0,0    0.1

1,1



Three numbers, 6 tiles

0,0    0,1   0,2

1,1    1,2

2,2



Four numbers, 10 tiles

0,0    0,1   0,2   0,3

1,1    1,2   1,3

2,2    2,3

3,3


Five numbers, 15 tiles

0,0    0,1   0,2   0,3    0,4

1,1    1,2   1,3   1,4

2,2    2,3    2,4

3,3    3,4

4,4


Six numbers,  21 tiles

0,0    0,1   0,2   0,3    0,4    0,5

1,1    1,2   1,3   1,4    1.5

2,2    2,3    2,4   2.5

3,3    3,4    3,5

4,4    4,5

5,5   


Seven numbers, 28 tiles 

0,0    0,1   0,2   0,3    0,4    0,5   0,6

1,1    1,2   1,3   1,4    1.5    1,6

2,2    2,3    2,4   2.5    2,6

3,3    3,4    3,5   3,6

4,4    4,5    4,6

5,5     5,6

6,6 







BONUS

See the ""Math in Backgammon" in the bonus part of this book.


RETURN



Some people might assume that we need 42 tiles to capture the combinations because there are seven numbers or 7 x 7.  After all, they learned when analyzing moves in backgammon that there are 36 combinations for the six-sided dice.   


The difference becomes clear when you count the duplicated combinations.   Look at the combinations that are mirror reflections of each other:  1,6 is the same as 6,1.  When we add the duplications, we find that the number goes up to 49.

In the table showing seven numbers, we see there are seven tiles with doubles and 21 tiles with pairs of different numbers.  We could have 49 tiles, but 42 of them would have duplicates.  


This step-by-step analysis, looking at what is in front of us, is part of the benefits tat come to a class of students who are GUIDED in the use of dominoes.   WIthout questions, students might not know how to analyze a situation.


one way to teach this is to create a grid

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