TAIT'S GOLF MATCH PROBLEM 



103 



problem ; "When a golf-player is x holes 'up' and y 'to play,' in how many 

 ways may the game finish ? " The paper in which Tait considered the problem 

 is called a question of arrangement and probabilities. He first solved the 

 simpler question as to the number of ways the player who is x up and y to play 

 may win. Let this number of ways of winning be represented by P (x, y). 

 Then starting with P (x + i, y+ i), we see that at the first stage the player 

 may win, halve, or lose the next hole, and the number of possible ways of 

 winning will then be represented by P (x+2, y}, P (x+i, y}, and P (x, y) 

 respectively ; hence follows Tail's fundamental equation 



If then we construct a coordinate scheme with x measured horizontally 

 and y vertically downwards, and place in the position xy the number P (x, y), 

 we can at once pass by simple addition of three consecutive values of x for 

 any one value of y to the values for the next higher value of y. The 

 following is the scheme as far as jx=5- 



etc. 



etc. 



The zero positions are enclosed in the double lines ; and the meaning of 

 the entries to the left of the vertical lines is the number of ways in which the 

 player may lose. The unit values on the right and left flanks are determined 

 by the limiting conditions, which show that when x is greater than y, the game 

 is won, so that P (x, y} = i . Similarly, when x is not less than y, the player 

 cannot lose. Hence P(~x, y} = o. These considerations also explain why 

 the fundamental equation given above does not apply to the second last unit 

 on the right of each row. As an example, let a player be 2 up and 4 to play ; 

 he may win in 26 different ways. His opponent who is 2 down and 4 to play 

 may of course lose in the same number of ways. But the number of ways in 

 which the player who is 2 up may lose is only 5. These numbers 26 and 5 



