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George Lowther
George Lowther
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There is a simple proof that a game of Hex must have a winner, which implies the result you want.See here: Brouwer's Fixed Point Theorem and the Jordan Curve Theorem, Lemma 5.5. Take a look atThe Brouwer fixed point theorem and the Jordan Curve theorem follow from this.

This proof is based on the paper The Game of Hex and the Brouwer Fixed-Point Theorem (by David Gale. The American Mathematical Monthly, Vol. 86, No. 10. (Dec., 1979), pp. 818-827). The

Edit: Actually the reference shows that the Game of Hex always has a winner => Brouwer fixed pointFixed Point theorem and=> a pair of curves in the Jordan Curve theorem follow from thissquare joining opposite corners must intersect.

Also So it does use Brouwer's fixed point theorem, there is a shortbut gives an elementary proof here: Brouwer's Fixed Point Theorem and the Jordan Curve Theoremof it.

There is a simple proof that a game of Hex must have a winner, which implies the result you want. Take a look at The Game of Hex and the Brouwer Fixed-Point Theorem (by David Gale. The American Mathematical Monthly, Vol. 86, No. 10. (Dec., 1979), pp. 818-827). The Brouwer fixed point theorem and the Jordan Curve theorem follow from this.

Also, there is a short proof here: Brouwer's Fixed Point Theorem and the Jordan Curve Theorem.

There is a simple proof that a game of Hex must have a winner, which implies the result you want.See here: Brouwer's Fixed Point Theorem and the Jordan Curve Theorem, Lemma 5.5. The Brouwer fixed point theorem and the Jordan Curve theorem follow from this.

This proof is based on the paper The Game of Hex and the Brouwer Fixed-Point Theorem (by David Gale. The American Mathematical Monthly, Vol. 86, No. 10. (Dec., 1979), pp. 818-827).

Edit: Actually the reference shows that the Game of Hex always has a winner => Brouwer Fixed Point theorem => a pair of curves in the square joining opposite corners must intersect. So it does use Brouwer's fixed point theorem, but gives an elementary proof of it.

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George Lowther
George Lowther
  • 17.1k
  • 1
  • 66
  • 98

There is a simple proof that a game of HexHex must have a winner, which implies the result you want. Take a look at The Game of Hex and the Brouwer Fixed-Point Theorem (by David Gale. The American Mathematical Monthly, Vol. 86, No. 10. (Dec., 1979), pp. 818-827). The Brouwer fixed point theorem and the Jordan Curve theorem follow from this.

Also, there is a short proof here: Brouwer's Fixed Point Theorem and the Jordan Curve Theorem.

There is a simple proof that a game of Hex must have a winner, which implies the result you want. Take a look at The Game of Hex and the Brouwer Fixed-Point Theorem (by David Gale. The American Mathematical Monthly, Vol. 86, No. 10. (Dec., 1979), pp. 818-827). The Brouwer fixed point theorem and the Jordan Curve theorem follow from this.

Also, there is a short proof here: Brouwer's Fixed Point Theorem and the Jordan Curve Theorem.

There is a simple proof that a game of Hex must have a winner, which implies the result you want. Take a look at The Game of Hex and the Brouwer Fixed-Point Theorem (by David Gale. The American Mathematical Monthly, Vol. 86, No. 10. (Dec., 1979), pp. 818-827). The Brouwer fixed point theorem and the Jordan Curve theorem follow from this.

Also, there is a short proof here: Brouwer's Fixed Point Theorem and the Jordan Curve Theorem.

Source Link
George Lowther
George Lowther
  • 17.1k
  • 1
  • 66
  • 98

There is a simple proof that a game of Hex must have a winner, which implies the result you want. Take a look at The Game of Hex and the Brouwer Fixed-Point Theorem (by David Gale. The American Mathematical Monthly, Vol. 86, No. 10. (Dec., 1979), pp. 818-827). The Brouwer fixed point theorem and the Jordan Curve theorem follow from this.

Also, there is a short proof here: Brouwer's Fixed Point Theorem and the Jordan Curve Theorem.