Dominos

(thanks to the New Scientist Magazine)


I have a standard set of 28 dominos. I take some and arrange them into a rectangle. The rectangle has the property that no horizontal or vertical straight line can be drawn across it, which doesn't bisect at least one domino. I take this rectangle apart, add two more dominos and make another rectangle with the same property. How many dominos are in my second rectangle?

where does the line start? inside or outside the rectangle?

The line must be horitontal or vertical and start and finish outside the rectangle.

For instance

||=
=||

(where || is a vertical domino, and = is a horizontal domino)

... cannot be one of my rectangles because it can be divided neatly by both a horizontal cut:

||=
.....
=||

and a vertical one:

|| . =
= . ||

sorry , that was very confusing, only worked if = and || each denoted two dominos.

Hopefully this is clearer:

axoo
axax
ooax

where two adjacent pieces with the same letter compromise each domino.

can be cut like this:

ax oo
ax ax
oo ax

So it is not a possible rectangle.

Nice one - proof?

25 is possible:

AABAABBCCA
BCBCCAADDA
BCDDBCCBBC
DAACBDDAAC
DBBCAABBDD

27 is possible:

AABAABBCC
BCBCCAADD
BCDDBCCBB
DAACBDDAA
DBBCAABBD
AADDCCAAD

Edit: cannot be hidden.

Very good!