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If A picks B, B picks C and C picks A, that would all work out.

However, any one of the three know their own recipient of their own gift &, as that recipient cannot reciprocate without leaving the third with themselves as their respective selection, everybody knows who bought a gift for the other two in the group.

For example,

If A buys gift for B, B cannot buy for A, so B must have bought a gift for C &, hence, C must have bought a gift for A.

If A buys for C, C cannot buy for A, so must have bought for B.

If B buys for A, A cannot buy for B, so A must have bought for C, & C bought for B.

If B buys for C, C cannot buy for B, so C bought for A, & A bought for B.

If C buys for A, A cannot buy for C, so A bought for B, & B bought for C.

If C buys for B, B cannot buy for C, so B bought for A, & A bought for C.

It doesn't work with three people.

Yeah, you're right. If I know who my recipient is they can't have me picked, so I know they have the other person.

I was just about to get on to it...

If you had limit of say £20 each in mind, split in half. 6 numbers go into hat, you each pick out two numbers. Buy pressies, place your number onto each pressie. On the day of giving, you each pick one number out only. You may even get the thing you bought yourself. The remaining three presents regardless if anyone ended up with own gift, the other three you give to local charity etc.