## Task 2 - 270922

In a "Hungarian domino game" with the numbers $0,1, \ldots, 9$, each tile is divided into two halves, each half bearing one of the numbers.

In a game, all combinations of any two of the numbers $0,1, \ldots, 9$ appear exactly once (including those where the same number appears on both halves of a tile).

A "chain" is formed by placing several tiles next to each other 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 "Hungarian domino game"!

b) Determine the maximum number of such tiles from a game that can form a closed chain!
a) We assume w.l.o.g. that the first digit on a domino tile is greater than or equal to the second digit. Then there is exactly one domino tile with the first digit 0 (namely 0-0, exactly two with the first digit 1 (namely 1-0 and 1-1), ..., exactly ten domino tiles with the first digit 9 (namely 9-0, 9-1, ..., 9-9). In total, there are thus $1+2+\cdots+10=55$ different domino tiles.

b) Each digit appears on these 55 domino tiles a total of eleven times, namely nine times with each other digit and twice on the domino tile that has the respective digit on both sides. However, each digit appears an even number of times in a closed chain, namely in pairs on adjacent halves of two neighboring domino tiles. Therefore, in a closed chain, each digit can appear at most ten times, so that at most 50 tiles can be used in the closed chain.

A chain can also be obtained as follows:

$0-0,0-1,1-1,1-2,2-2,2-3,3-3,3-4,4-4,4-5,5-5,5-6,6-6,6-7,7-7,7-8,8-8,8-9,9-9,9-0,0-2,2-4,4-6,6-8,8-1,1-3,3-5,5-9,9-7,7-0,0-3,3-6,6-9,9-2,2-5,5-1,1-7,7-4,4-8,8-0,0-4,4-1,1-9,9-3,3-7,7-5,5-8,8-2$, 2-6, 6-0

Here, the domino tiles 0-5, 1-6, 2-7, 3-8, and 4-9 were not used, and all others were used exactly once.