Gem 16: The "Who is the Engineer?" Puzzle Solution
By Moshe Shweiger

	The "Who is the Engineer?" Puzzle
1..	Smith, Jones, and Robinson are the engineer, brakeman, and fireman on a train, but not necessarily in that order. Riding the train are three passengers with the same three surnames to be identified in the following premises by a "Mr." before their names.
2..	Mr. Robinson lives in Los Angeles.
3..	The brakeman lives in Omaha.
4..	Mr. Jones long ago forgot all the algebra he learned in high school.
5..	The passenger whose name is the same as the brakeman's lives in Chicago.
6..	The brakeman and one of the passengers, a distinguished mathematical physicist, attend the same church.
7..	Smith beat the fireman at billiards.
	Who is the engineer?

Here is a short, concise solution using matrix notation:

Solution to "Who is the Engineer?"
We see that the passengers live in different cities since one lives in Los Angeles (2), another lives in Chicago (5), and the physicist lives in Omaha (3) since he lives in the brakeman's city (6). We construct 3 matrices A, B, and C. Each matrix is 3 rows by 3 columns, and we will refer to their cells in the usual matrix notation like A11, B13, C23, etc. A "1" in a cell is a "match" and a "0" is a "mismatch".
Matrix A Engineer Brakeman Fireman Smith A11 A12 A13     0 Jones A21 A22       1 A23 Robinson A31 A32 A33	From 7, A13 = 0.
Matrix B Los Angeles Omaha Chicago Mr. Smith B11 B12   1 B13 Mr. Jones B21 B22   0 B23   1 Mr. Robinson B31   1 B32 B33	From 2, B31 = 1. From 3, C22 = 1. From 4, Mr. Jones is not the Physicist. From 6, the Physicist lives where the Brakeman lives, which is Omaha.
Matrix C Los Angeles Omaha Chicago Engineer C11 C12 C13 Brakeman C21 C22   1 C23 Fireman C31 C32 C33	This implies that Mr. Jones doesn't live in Omaha; the Physicist does. So B22 = 0. B31 and B22 tell us that Mr. Jones must live in Chicago, so B23 = 1.
Checking Matrix B we see that Mr. Smith cannot live in Los Angeles or Chicago, so Mr. Smith lives in Omaha and B12 = 1, and of course, he, Mr. Smith, is the Physicist. Mr. Jones lives in Chicago, and 5 tells us that the Brakeman's name is Jones, so A22 = 1. Smith is not the Fireman, and he is not the Brakeman (Jones is), so he must be the Engineer.
The answer to the puzzle's question "Who is the engineer?": Smith is the engineer. (SOLVED)

Here is another solution using a simple list of clues:

	We list all the clues to solve the mystery of "Who is the engineer?" including the first seven given in the puzzle statement above: (The clues basis for each new clue is included in parentheses.)
1..	Smith, Jones, and Robinson are the engineer, brakeman, and fireman on a train, but not necessarily in that order. Riding the train are three passengers with the same three surnames to be identified in the following premises by a "Mr." before their names.
2..	Mr. Robinson lives in Los Angeles.
3..	The brakeman lives in Omaha.
4..	Mr. Jones long ago forgot all the algebra he learned in high school.
5..	The passenger whose name is the same as the brakeman's lives in Chicago.
6..	The brakeman and one of the passengers, a distinguished mathematical physicist, attend the same church.
7..	Smith beat the fireman at billiards.
8..	The passengers live in different cities. - They live in Los Angeles (2), Chicago (5), and Omaha (3 since the physicist lives in the brakeman's city of Omaha from 6).
9..	Mr. Jones is not the physicist (4) and does not live in Omaha (8).
10..	Mr. Jones lives in Chicago (9 and 2).
11..	Mr. Smith lives in Omaha (2 and 10).
12..	Mr. Smith is the physicist (3 and 6).
13..	The brakeman's name is Jones (5 and 10).
14..	Smith is not the fireman (7) nor the brakeman (13), and so Smith is the engineer. (SOLVED)

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