I read Turing had some algorithm or method he could use to play chess 'against' this pen and paper algorithm assigning certain numerical values to each piece. Is this true? Is there a pen and paper algorithm one can use to play a kind of chess with oneself?
Yes, he played a game against Alick Glennie, actually he created an algorithm based on mathematical calculations. Then he tried his algorithm using pen and paper to do calculations.
[fen ""] [White "Alan Turing"] [Black "Alick Glennie"] 1.e4 e5 2.Nc3 Nf6 3.d4 Bb4 4.Nf3 d6 5.Bd2 Nc6 6.d5 Nd4 7.h4 Bg4 8.a4 Nxf3+ 9.gxf3 Bh5 10.Bb5+ c6 11.dxc6 O-O 12.cxb7 Rb8 13.Ba6 Qa5 14.Qe2 Nd7 15.Rg1 Nc5 16.Rg5 Bg6 17.Bb5 Nxb7 18.O-O-O Nc5 19.Bc6 Rfc8 20.Bd5 Bxc3 21.Bxc3 Qxa4 22.Kd2 Ne6 23.Rg4 Nd4 24.Qd3 Nb5 25.Bb3 Qa6 26.Bc4 Bh5 27.Rg3 Qa4 28.Bxb5 Qxb5 29.Qxd6 Rd8 0-1
The main idea of algorithm is here:
Mobility: For the queen, rooks, bishops, add the square roots of the number of moves that the piece can make, counting a capture as two moves.
Piece safety: For the rooks, bishops and knights add 1 point if there is one defender and 1.5 if there is more than one
King mobility: For the king use the same method of scoring as for the piece, but do not count castling
King safety: Deduct points for the king's vulnerability, defined as the number of moves that a queen could make were it on the square of the king
Castling: add 1 point if castling is still legally possible after this move. Add another point if castling is immediately possible or if a castling move has just been made
Pawn credit: score .2 points for each rank advanced and .3 points for each pawn defended by one or more non pawns
Check and mate threats. Score 1 point for the threat of mate, and .5 points for a checkmaterial values used to each of the pieces were:pawn =1, knight=3, bishop=3.5 rook =5, queen=10
Yes, Turing created a "Paper Machine" along with David Champernowne in 1948 that they called "Turochamp". As far as I know he only played one recorded game with the system, against Alick Glennie (full game found here). There was only one recorded game against the Paper Machine because it took a very long time. Calculating one move without an electronic computer could take up to an hour.
You can read more about Turochamp here.
I have not found any source code for running Turochamp yourself, but I"m sure someone out there has coded it. For the 100th anniversary of Turing's birth Kasparov played a game against the Paper Machine, notes here.
Chessbase has made an implementation of Turing's engine available at the bottom of this post.