244 MISCELLANEOUS PROBLEMS [CH. XI 



the smallest positive number (exclusive of zero) which must 

 be added to p to make the sum a multiple of 3, and c is that 

 multiple. 



A couple of illustrations will make this clear. Suppose 

 we wish the card to come out 22nd from the top, therefore 

 22 = 9c — 36 + a. The smallest number which must be sub- 

 tracted from 22 to leave a multiple of 3 is 1, therefore a = 1. 

 Hence 22 = 9c -36 + 1, therefore 7 = 3c -b. The smallest 

 number which must be added to 7 to make a multiple of 3 is 2, 

 therefore 6 = 2. Hence 7 = 3c — 2, therefore c = 3. Thus a = 1, 

 b = 2, c = 3. 



Again, suppose the card is to come out 21st. Hence 

 21 = 9c — 36 + a. Therefore a is the smallest number which 

 subtracted from 21 makes a multiple of 3, therefore a = 3. 

 Hence 6 = 3c — 6. Therefore b is the smallest number which 

 added to 6 makes a multiple of 3, therefore 6 = 3. Hence 

 9 = 3c, therefore c = 3. Thus a = 3, 6 = 3, c = 3. 



If any difficulty is experienced in this work, we can proceed 

 thus. Let a=x + 1, 6 = 3 — y, c = z+ 1; then x, y, z may have 

 only the values 0, 1, or 2. In this case Gergonne's equation 

 takes the form 9z + 3y + x = n — 1. Hence, if w. — 1 is expressed 

 in the ternary scale of notation, x, y, z will be determined, and 

 therefore a, b, c will be known. 



The rule in the case of a pack of m m cards is exactly similar. 

 We want to make the card come out in a given place. Hence, 

 in Gergonne's formula, we are given n and we have to find 

 a,b, ..., k. We can effect this by dividing n continually by m, 

 with the convention that the remainders are to be alternately 

 positive and negative and that their numerical values are to be 

 not greater than m or less than unity. 



An analogous theorem with a pack of Im cards can be con- 

 structed. C. T. Hudson and L. E. Dickson* have discussed the 

 general case where such a pack is dealt n times, each time into 

 I piles of m cards ; and they have shown how the piles must be 



* Educational Times Reprints, 1868, vol. ix, pp. 89 — 91 ; and Bulletin of 

 the American Mathematical Society, New York, April, 1895, vol. I, pp. 184 — 

 186. 



