##
Common Letters

Three people (A, B, C) are each given two colors, for a total of six
colors (blue, green, orange, purple, red, and yellow.) In comparing
the colors they received, B makes the following observation:
"The number of common letters between my two colors is one less than
the number of A's common letters and one more than the number of
C's common letters."

Find the colors each person received. Is there more than one answer?

(For this discussion, dual letters are only counted once. For example,
*yellow* and *purple* have only two common letters: *e*
and *l*.)

Source: Original.

Solutions were received from:
Arturo Pascalin,
Denis Borris,
Robert T. McQuaid,
Carlos Rivera,
and
Larry Baum.
Robert McQuaid's solution follows:
Below is a matrix showing the number of common letters
between each pair of colors:
*blue
1 *green
1 4 *orange
3 2 2 *purple
1 2 2 2 *red
2 1 2 2 1 *yellow
Here the color attached to the * applies to each entry to
the left of the * and directly below it. For example, the
3 in the left column is under *blue and left of *purple,
so that entry refers the the three common letters between
blue and purple.
Since there are only 4 distinct values in the matrix,
the number of common letters for A B C is either
4 3 2 or 3 2 1. Simple examination discloses one solution:
[Larry Baum's analysis insered here - KD]
So either A=4 (orange/green) or 3 (blue/purple)
If A=4, then B=3 (blue/purple), which leaves (red/yellow)=1 for C, but
we would need C=2.
So A=3 (blue/purple). Then C=1 must be either red/yellow or green/yellow
leaving B= orange/green(4) or orange/red(2). Only the latter works.
A 3 purple/blue
B 2 orange/red
C 1 yellow/green

Mail to Ken