250 MISCELLANEOUS PROBLEMS [CH. XII 



digits form a repeating decimal, as indicated in the following 

 work, where a bar has been put above the repeating digits. It 

 is required to restore the working*. This problem is remarkable 

 from the fact that not a single digit is given explicitly. 



) ( • - • 



The answer is that the divisor is 667334 and the dividend is 

 7752341. 



Here are three additional examples of arithmetical restora- 

 tions f. The solutions are lengthy and involve much empirical 

 work. 



(iii) The first of these Berwick questions is as follows. In 

 the following division sum all the digits, except the seven "7's" 

 shown, have been erased : each missing digit may be 1, 2, 3, 4, 

 5, 6, 7, 8, 9, or (except in the first digit of a line) 0. Observe 



* American Mathematical Monthly, 1921, vol. xxviii, p. 61. 



t All are due to W. E. H. Berwick. The " 7 " problem appeared in the School 

 World, July and October 1906, vol. viii, pp. 280, 320; the "4" problem appeared 

 in the Mathematical Gazette, 1920, vol. x, pp. 43, 359 — 360 ; the " 5 " problem in 

 the same paper, vol. x, p. 361, vol. xi, p. 8. 



