The great Risk tournament is organized again. This year there are 1000 contestants. Each game in the tournament is played with 4 people. After each game, there will be one winner which goes to the next round. In the next round again groups of 4 players are formed. If one does not get an exact multiple of 4 in this way, some of the players in this round are selected randomly. These players can go to the following round automatically. In the end there will be one winner. How many games will be played?

Suppose: a = b

It then follows that
a * a = a * b
a^2 = a * b
a^2 - b^2 = a * b - b^2
(a - b) * (a + b) = (a - b) * b
a + b = b
a + a = a
2 * a = 1 * a
2 = 1

Is this derivation really correct? Or is there somewhere a mistake? If so, where?

Somebody would like to have a new garden with 10 trees which have to stand in 5 rows of 4 trees each. How can he or she achieve this? After the new garden has been layed out, neigbor Sym becomes enthousiastic. He says: "I also would like to have a new garden with trees. But it should have 12 trees which have to be planted in 6 rows of 4 trees each. Furthermore, my garden should be symmetric." How will his garden look like? And can you tell something more about Sym?