A blindfolded performer moves a knight to every square of a chessboard, landing on no square twice,...

competition/games/chess/knight.tour.any.start

Problem

A blindfolded performer moves a knight to every square of a chessboard,
landing on no square twice, starting from a square chosen by the
audience.  How can he do it without memorizing 64 different tours?

Once about 30 - 35 years ago, I watched a guy do a stunt in front of an
audience on television.  He had a giant chess board set up, tilted on a
stand, and after explaining how knights move in chess, he announced
that he could move a knight to every square on the chess board without
landing on any square twice.  He even let a member of the audience pick
his starting square.  He wore a blindfold and he had a girl, sort of
like Vanna White, place a marker over each square as he called out its
number.

He pretended to be confused at a couple of points, but he didn't fool
me.  I figured that he had the solution memorized.  The question was,
how could he let a member of the audience pick his starting square?
Did he memorize 64 different solutions?

-- S. Macaw <smacaw@ix.netcom.com>, July 1995

Show solution

He needs only one tour, provided it is a closed (re-entrant) tour: one
whose last square is a knight's move from its first.  Such a tour is a
single cycle through all 64 squares, so it can be started anywhere.
Whatever square the audience picks, he finds its number in his one
memorized tour and calls the squares in order from there, going on
from 64 to 1 and stopping when he has called 64 squares.  He does not
even need to remember where the cycle was "started".

Here is one closed tour, found and checked by a short C++ search
(each number is a knight's move from the next, and 64 (c2) is a
knight's move from 1 (a1)):

  8   22  7 44 39 24  9 28 59
  7   43 40 23  8 45 60 25 10
  6    6 21 42 61 38 27 58 29
  5   41 62 37 46 57 54 11 26
  4   20  5 56 53 36 47 30 51
  3   63  2 35 48 55 52 15 12
  2    4 19 64 33 14 17 50 31
  1    1 34  3 18 49 32 13 16
       a  b  c  d  e  f  g  h

If the audience picks, say, e4 (number 36), he calls 36, 37, ..., 64,
then 1, 2, ..., 35.

A performer can make the one tour easier to remember by numbering the
squares 1 to 64 and learning the tour as a list of square numbers, or
by choosing a symmetric tour.  Closed tours are plentiful:
Euler (1759) already gave methods for making them, and there are about
1.3 * 10^13 of them on the 8x8 board (B. McKay, 1997).
-- Claude (rec.puzzles Oracle, 2026-11-02)

[ Editor: the material collected earlier follows. ]

[ Editor: the poster added the following; see also
competition/games/chess/knight.tour. ]

Incidently, I wrote a QuickBasic program that methodically searches out
solutions to this puzzle.  Just think, you could memorize the solution
and be a big hit at the next party!  If anyone's interested, just say
so.

Visit Strawberry Macaw's Puzzle Page at
http://wworks.com/~macaw

-- S. Macaw <smacaw@ix.netcom.com>, July 1995