Question

Consider the following 11 triples: ABE ACE ADE BCD ABF ACF ADF EFG ABG ACG ADG Prove that no matter how the 7 letters A through G are colored red or blue, the letters of one of these 11 triples will be either all red or all blue.

Answer 1

consider the numbers are colored alternate red and blue.

from the arrangement it can be seen that the pairing is such that it starts from 1st letter and has a triples of all odd numbers (A=1,B=2,C=3,D=4,E=5,F=6,G=7) i.e.ACE.

therefore all colours will be same in this arrangement.

if we consider arrangement of 2 pairs as -

ABCDEFG

RRBBRRB

respectively, then also we will get a triple of all same colours.

The pairing is such divide so that the first letters come first and second and third one is from last 3 letters(E,F,G).

hence if we try any pair of colours we will always get a triple with all same colour.