CH. XII] MISCELLANEOUS PROBLEMS 253 



that b cannot be 6. A few trials now show that the question 

 arose from the division of 19,775 by 35. 



It is possible to frame digit restoration examples of a mixed 

 character involving the difficulties of all the examples given 

 above, and to increase the difficulty by expressing them in a 

 non-denary scale of notation. But such elaborations do not add 

 to the interest of the questions. 



Calendar Problems. The formulae given by Gauss and 

 Zeller, which I quoted in former editions of this work, serve to 

 solve all questions likely to occur about dates, days of the week, 

 Easter, etc. Here are two easy but elegant questions on the 

 Gregorian Calendar of a somewhat different nature. 



The first is as follows : it is due I believe to E. Fourrey. In 

 the century and a half between 1725 and 1875 the French fought 

 and won a certain battle on 22 April of one year, and 4382 days 

 later, also on 22 April, they gained another victory. The sum 

 of the digits of the years is 4©. Find the dates of the battles. 



To solve it we notice that 4382 = 12 x 365 + 2. Hence the 

 date of the second battle was 12 years after that of the first 

 battle ; but only two leap years had intervened, and therefore 

 the year 1800 must be within the limiting dates. Thus 1788 



and 1800, 1789 and 1801, 1800 and 1812, are the only 



possible years. Of the years thus suggested 1796 and 1808 

 alone give 40 as the sum of their digits. Hence the battles 

 were fought on the 22 April, 1796 (Mondovi under Napoleon) 

 and 22 April, 1808 (Eckmuhl under Davoust). 



The other of these questions is to show that the first or last 

 day of every alternate century must be a Monday. This follows 

 from knowing any one assigned date, and the fact that the 

 Gregorian cycle is completed in 400 years. 



I proceed next to mention as additional Geometrical Recrea- 

 tions, the Five Disc Problem, Tesselation with Super-Dominoes, 

 Geometrical Dissections, the Window-Reader Problem, and 

 Compass Problems. 



The Five Disc Problem. The problem of completely 

 covering a fixed red circular space by placing over it, one at a 



