Imagine an array of 3 x 7 lights. Lights can be on or off. Show that no matter which lights you turn on or off, you will always be able to find four lights forming the corner of a rectangle, either all on or all off.

Again consider 3 columns and 7 rows.

Column 1 must have a majority of either on or off lights.  Choose whichever state that is.  It has at least 4 occurrences in column 1.

In column 2, if at least 2 of these 4 rows match column 1, you have a rectangle. If only 1 or 0 match, you have at least 3 that don't match (i.e., are all the opposite).

In column 3, these rows must have at least two that match each other.  They will also both match either column 2 or column 1 in that row.

Posted by Charlie on 2004-09-14 13:37:55