Rahul Mehta Gandhinagar LsCandidate

An excellent logic puzzle.
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(nothing to do with politics economics)
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Puzzle-1 :
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(1a) There are two players, B and C.
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(1b) The host will put a number on their head - 0 or 1.
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(1c) There can be repetitions - so both may get 0 and both may get 1.
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(1d) A player can see ONLY the other player's number and cant see eyes or hands (so cant use eyes or hands to pass information).
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(1e) Both are given a slip of paper, on which they can make one and ONLY one guess. (no talking , only writing, and writing ONLY once and they cant see others' answers)
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And each player has to guess the number on his own head.
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If anyone , B or C, guesses his number right, then both win. Or both loose. So pls note --- BOTH need not guess the number on their heads correctly. ONLY ONE needs to guess correctly to win the game.
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The players can discuss and make a strategy before the game starts.
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So what strategy will enable them to win in all 4 cases?
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(such a strategy exists)\
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Puzzle-2 : same as above with 4 players.

(2a) There are 4 players, B, C D, and E.
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(2b) The host will put a number on their head - 0 , 1 , 2 or 3.
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(2c) There can be repetitions - so all may get 0 or some combination like (0 , 1, 1 , 2) etc. So there are 4 * 4 * 4 * 4 = 256 possibilities
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(2d) Each player can see ONLY the other players' numbers and cant see eyes or hands (so cant use eyes or hands to pass information).
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(2e) All 4 are are given a slip of paper. On slip, after looking at others' numbers, they have to write their guess on number on their own head. They can write one and ONLY one guess.
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If anyone , B or C or D or E,, guesses his number right, then group wins. Or group looses. So pls note --- all 4 need not guess the numbers on their heads correctly. ONLY ONE needs to guess correctly to for group to win the game.
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The players can discuss and make a strategy before the game starts.
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So what strategy will enable them to win in all 256 cases?
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(such a strategy exists)\