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Exactly what it says on the tin, as the British say:

In Hand and Brain chess, is the stronger player generally preferred to be the hand or the brain?

And secondarily:

Is the answer to the previous question affected at all by the particular relative strengths of the players (as in, is the answer for a team involving Magnus Carlsen and I (probably around 1200) substantially different than that for team involving a 1600 and an 1800)?

I can think of good reasons for both: the stronger player being the hand might be nice because they could select the best move under some constraints, and thus only very rarely make an outright blunder (generally, the only way they could play an outright blunder (assuming they're strong enough to pretty reliably avoid blunders) would be is a piece was chosen that had NO good moves, which I think is probably pretty rare). On the other hand, a stronger player being the brain might be preferred, as they could narrow down a decision tree for the lesser player in the hopes that the best move might become easier to see without such a large tree to analyze.

To be clear, Hand and Brain chess is a type of team chess in which two players play each side. One (the "Brain") is only allowed to pick a piece type to be moved, and the other (the "Hand") must move that piece type without knowledge of what the first player intended. For example, if the "Brain" says "pawn", then the "Hand" must move one of the pawns. If the "Brain" says "king", then the "Hand" must move the king (includes castling).

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Not a definitive answer, but I'm posting my comment based on the upvotes it received.

It depends. GMs have different opinions.

GM Hikaru Nakamura has said it really depends on the situation. Additionally, he has commented that when the Brain, he often suggests moves that aren't correct because he is factoring in the likely Hand response to his move.

GM Aman Hambleton stated the stronger ELO player should be the Hand generally because they will find good moves in most cases, and be able to identify a plan for a position much easier.

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    Do you have links or citations for those GM statements? – Pureferret May 6 at 10:05
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    @Pureferret Great point. Both statements were made on their Twitch streams...it will take a lot of time to go through their collective 5 Youtube channels and their Twitch VODs to find these clips. If I find them, I'll edit to include the links as citations. – SecretAgentMan May 6 at 12:47
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The following is just my personal take on this, based on never having played hand-brain chess and on my extremely limited understanding of chess as a very weak player (chess.com ~1300 blitz, maybe ~1400 on a good day in classical). So take this with a grain of salt.

I would think that if both players are relatively strong but one is clearly stronger (say an IM and a GM), the weaker of the two should be the hand and the stronger the brain. This is because the GM can likely trust that the IM would make the right move when given the piece to move. The IM would have a good enough understanding of the position to make the best move of the chosen piece.

If one player is head and shoulders above the other, say someone like me and a strong rated player, then the weaker player should probably be the brain. I figure if I tell the strong player to move a piece, they'll manage to make the best of it. If I am the hand, I am more likely to make a mistake even if the stronger player gives me the right piece while if they are the hand, they would have a better chance of finding a good move even if I don't find the best move and suggest the best piece.

If both players are relatively weak, I expect it makes no difference at all who is hand and who is brain.

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