Problem of the Week Problem C Domi Knows

A domino tile is a rectangular tile with a line dividing its face into two square ends. Each end is marked with a number of dots (also called pips) or is blank.

The first domino shown below is a $$[3,5]$$ domino, since there are $$3$$ pips on its left end and $$5$$ pips on its right end. The second domino shown below is a $$[0,3]$$ domino, since there are $$0$$ pips on its left end and $$3$$ pips on its right end. The third domino shown below is a $$[4,4]$$ domino, since there are $$4$$ pips on its left end and $$4$$ pips on its right end.

We can also rotate the domino tiles. The first domino shown below is a $$[5,3]$$ domino, since there are $$5$$ pips on its left end and $$3$$ pips on its right end. However, since this tile can be obtained by rotating the $$[3,5]$$ tile, $$[5,3]$$ and $$[3,5]$$ represent the same domino. Similarly, the second domino shown below is a $$[3,0]$$ domino. Again, note that $$[3,0]$$ and $$[0,3]$$ represent the same domino.

A $$2$$-set of dominoes contains all possible tiles with the number of pips on any end ranging from $$0$$ to $$2$$, with no two dominoes being the same. A $$2$$-set of dominoes has the following six tiles: $$[0,0]$$, $$[0,1]$$, $$[0,2]$$, $$[1,1]$$, $$[1,2]$$, $$[2,2]$$. Notice that the three dominoes $$[1,0]$$, $$[2,0]$$, and $$[2,1]$$ are not listed because they are the same as the three dominoes $$[0,1]$$, $$[0,2]$$, and $$[1,2]$$.

Similarly, a $$12$$-set of dominoes contains all possible tiles with the number of pips on any end ranging from $$0$$ to $$12$$, with no two dominoes being the same.

Domi purchased a $$12$$-set of dominoes. How many tiles are in the set?

Theme: Number Sense