Chap. XX] PROPERTIES OF THE DiGITS OF NUMBERS. 465 
SeveraP^" gave 9'w!+n+l= 1 . . .1 for n^9, with generalization to any 
base. 
E. J. Moulton^^ found the number of positive integers with r+1 digits 
fewer than p of which are unity (or zero). L. O'Shaughnessy^^ found the 
number of positive integers < 10* which contain the digit 9 exactly r times. 
Books^^ on mathematical recreations may be consulted. 
F. A. Halliday^^ considered numbers N formed by annexing the digits 
of B to the right of A, such that N= (A+B^, as for 81= (8 + 1)1 Set 
N=A-W+B. Then A{W-1)= iA+B)(A-}-B-l), so that it is a ques- 
tion of the factors of lO'' — 1. 
*J. J. Osana^^ discussed numbers of two and three digits. 
E. Gelin^^ listed 450 problems, many being on digits. 
*^"L'intermediaire des math., 2.5, 1918, 44-5. 
"Amer. Math. Monthly, 24, 1917, 340-1. 
8«/6id., 25, 1918, 27, 
*^E. Lucas, Arithmetique amusante, 1895. E. Fourrey, Recreations Arithmetiques, 1899. 
W. F. White, Scrap-Book of Elem. Math., etc. 
90Math. Quest, and Solutions, 3, 1917, 70-3. 
siRevista Soc. Mat. Espafiola, 5, 1916, 156-160. 
92Mathesis, (2), 6, 1896, Suppl. of 34 pp. 
