You have 36 playing cards that are to be arranged in a 6x6 array. Each card is to be placed initially face up into one of the 36 positions allocated, not necessarily adjacent to any previously placed card.

When a newly placed card, though, is in fact adjacent edge-to-edge (not diagonally), to a previously placed card, the previously placed card is flipped: face-up to face-down, or vice versa. A card may then be flipped more than once as additional cards are placed next to it. And a newly placed card may cause this to happen in more than one previously placed card, if more than one occupy orthogonally adjacent positions.

The goal is to have the completed 6x6 array consist of all face-up cards.

What is the placement order to make this happen? Or prove it's impossible.

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Based on "Not over easy", Enigma No. 1767, by Bob Walker, New Scientist, 21 September 2013, page 30.

The following problem is identical:

Place the numbers 1 to 36 in a 6x6 array such that each number is smaller than an even number of adjacent numbers.

One of many solutions is

01 04 09 16 25 36
03 08 15 24 35 30
07 14 23 34 29 20
13 22 33 28 19 12
21 32 27 18 11 06
31 26 17 10 05 02
The upper left corner numbers are bordered by larger numbers only to the right and below.
The lower right corner numbers are bordered by larger numbers only to the left and above.
The diagonal has no larger adjacent numbers.

 [The numbers become the order to place the cards.]

Posted by Jer on 2013-11-20 14:02:18