## Task 2 - 270912

In a domino game with the numbers $0,1, \ldots, 6$, each tile is divided into two halves, each half bearing one of the numbers. In a domino set, all combinations of two of the numbers $0,1, \ldots, 6$ appear exactly once (including those where the same number appears on both halves of a tile).

A "chain" is formed when multiple tiles are placed next to each other in a sequence such that adjacent halves of neighboring tiles always bear the same number (domino game rule).

A chain is called "closed" if the two tile halves at the free ends of the chain also bear the same number (so that the chain, if it consists of enough tiles, can be led back to its beginning and closed there).

a) Determine the number of all tiles belonging to a domino set!

b) Determine the largest number of such tiles from a domino set that can form a closed chain!
a) There are exactly 7 stones where the same number appears on both halves of the stone.

To describe the other stones, one can choose one of the 7 numbers for the first field and one of the 6 other numbers for the second field. In this way, each stone of the mentioned type is counted exactly twice. The number of stones thus amounts to $\frac{7 \cdot 6}{2}=21$.

The total number of stones in a domino set is therefore $7+21=28$.

b) One can form a closed chain with all the stones of a domino set, for example:

$00 / 01 / 11 / 12 / 22 / 23 / 33 / 34 / 44 / 45 / 55 / 56 / 66 / 61 / 13 / 35 / 51 / 14 / 46 / 62 / 24 /$
$40 / 02 / 25 / 50 / 03 / 36 / 60$

The largest number for a closed chain is therefore 28.