strategy - Leveling up and Getting Items!

You are playing an online RPG game with 9 friends of yours (10 people in total) and in the game, there is a 2-player dungeon where you level up every time you enter and complete it. Only up to two players are allowed inside it at a time. At the beginning of the game, everybody is level 1 as usual. Before the game released, there was a beta and you know how to level up fast from it, but there are some tricks you discovered.

• You cannot complete the dungeon alone.


• After completing this 2-player dungeon with your friend, you and your friend will get 1 level.



• If you play the dungeon with someone exactly one level below you, you will get an extra item at the end of the dungeon, but you cannot enter the dungeon with a friend more than one level below you.


• If you try to enter the dungeon with the same person again, no one will get any experience to level up or item.


• You can share the items later, they are not soulbound!

What is the maximum number of extra items you will get in total as a group from this dungeon?

Answer

Let's start low: 40.

Let's name the players A,B,C,D,E,F,G,H,I,J and define "n-next" of a player to be the player who is positioned n places afterwards, with round wrapping. Eg, the 2-next of A is C, and the 3-next of J is C.

1. First, AB go in to become level 2.


2. Then, for eight rounds, the player who was 1-next of his partner in the previous round goes in along with their 1-next (ie: BC,CD,DE...IJ). This yields 8 EI(Extra Items),as each time one level 1 and one level 2 go in. Now A,J are level 2 and the rest are level 3.


3. Each time there are 2 players with a specific level and all the others have one level more, there is always (checked practically but can't prove) a set of 10 pairs such that a) All pairs have the same "distance" between players, b) there is an order to go in so that, after the 10 pairs are finished, all the levels are 2 higher that the beginning 3) each pair produces an EI. Thus, for distance of 2,3 and 4, we have 30 EI.


4. Unfortunately, the same is not true for distance 5. You can only find 2 pairs here, the two low-levels and their corresponding 5-next.

This yields a total of 8+30+2=40 items.