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This short game (Queen's Pawn, Budapest Defence and ending in a Smothered Mate) illustrates the differences in method of recording chess moves. The version in Forsyth (recording the diagrammatic position rather than the moves preceding) is also viewable.
Board Position
Full Algebraic
Short Algebraic
1.d2-d4	 Ng8-f6
2.c2-c4	 e7-e5
3.d4xe5	 Nf6-g4
4.Ng1-f3 Nb8-c6
5.Bc1-f4 Bf6-b4+
6.Nb1-d2 Qd8-e7
7.a2-a3	 Ng4xe5
8.a3xb4? Ne5-d3#
1.d4	Nf6
2.c4	e5
3.dxe5	Ng4
4.Nf3	Nc6
5.Bf4	Bb4+
6.Nbd2	Qe7
7.a3	Ngxe5
8.axb4?	Nd3#
1.P-Q4	 Kt-KB3
2.P-QB4	 P-K4
3.PxP	 Kt-Kt5
4.Kt-KB3 Kt-QB3
5.B-B4	 B-Kt5 ch
6.QKt-Q2 Q-K2
7.P-QR3	 KKtxKP
8.PxB?	 Kt-Q6 mate

Basics Notation 3 - Forsyth Basics - Notation Overview