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There are 7 young men. All are excellent basketball players. They each started to 1v1 another player on their team.

They all played and nobody was left out. Everyone was present and none left the practice. Why was player #7 not left out? Who did he play with?

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There was 7 young men. All excellent basketball players.

So good in fact, that one of them could have played a 1 vs. 1 against another player who was already playing 1 vs. 1 against a different player. But that is not what happened.

They each started to 1vs1 another player on their team.

We will come back to this.

They all played and nobody was left out. Everyone was present and none left the practice. Why was player#7 not left out? Who did he play with?

#7 played against #4.

But in case of the odd man out, he played against #8. See there were 7 young men, but this was not the entirety of the team, there was an eight man. After all, in a game there are 5 players plus three in reserve.

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This answer works due to the ambiguity of the description of the scenario:

"There was 7 young men."

"They each started to 1vs1 another player on their team."

So

The basketball team contained 8 people, but only 7 of the 8 were young.
Player 1 played against player 8 and etc.
Nobody was left out and nobody left, so this fits the criteria.

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You said "they each started to 1vs1 another player on their team".

That means Player 1 played one-on-one against Player 7.

Then Player 2 played one-on-one against Player 7.

Then Player 3 played one-on-one against Player 7.

Then Player 4 played one-on-one against Player 7.

Then Player 5 played one-on-one against Player 7.

Then Player 6 played one-on-one against Player 7.

No one was left out. Player 7 played against the other 6 players.

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    $\begingroup$ They did, after all, have only one ball. $\endgroup$ – Michael Nov 15 '14 at 3:25

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