pints. A man who would involve the monastery in a fraction of fifty-eight decimals deserved severe punishment. 46.--_The Riddle of the Crusaders._ The correct answer is that there would have been 602,176 Crusaders, who could form themselves into a square 776 by 776; and after the stranger joined their ranks, they could form 113 squares of 5,329 men--that is, 73 by 73. Or 113 × 73^2 - 1 = 776^2. This is a particular case of the so-called "Pellian Equation," respecting which see _A. in M._, p. 164. 47.--_The Riddle of St. Edmondsbury._ The reader is aware that there are prime numbers and composite whole numbers. Now, 1,111,111 cannot be a prime number, because if it were the only possible answers would be those proposed by Brother Benjamin and rejected by Father Peter. Also it cannot have more than two factors, or the answer would be indeterminate. As a matter of fact, 1,111,111 equals 239 x 4649 (both primes), and since each cat killed more mice than there were cats, the answer must be 239 cats. See also the Introduction, p. 18. Treated generally, this problem consists in finding the factors, if any, of numbers of the form (10^_n_ - 1)/9. Lucas, in his _L'Arithmétique Amusante_, gives a number of curious tables which he obtained from an arithmetical treatise, called the _Talkhys_, by Ibn Albanna, an Arabian mathematician and astronomer of the first half of the thirteenth century. In the Paris National Library are several manuscripts dealing with the _Talkhys_, and a commentary by Alkalaçadi, who died in 1486. Among the tables given by Lucas is one giving all the factors of numbers of the above form up to _n_ = 18. It seems almost inconceivable that Arabians of that date could find the factors where _n_ = 17, as given in my Introduction. But I read Lucas as stating that they are given in _Talkhys_, though an eminent mathematician reads him differently, and suggests to me that they were discovered by Lucas himself. This can, of course, be settled by an examination of _Talkhys_, but this has not been possible during the war. The difficulty lies wholly with those cases where _n_ is a prime number. If _n_ = 2, we get the prime 11. The factors when _n_ = 3, 5, 11, and 13 are respectively (3 . 37), (41 . 271), (21,649 . 513,239), and (53 . 79 . 265371653). I have given in these pages the factors where _n_ = 7 and 17. The factors when _n_= 19, 23, and 37 are unknown, if there are any.[B] When _n_ = 29, the factors are (3,191 . 16,763 . 43,037. 62,003 . 77,843,839,397); when _n_ = 31, one factor is 2,791; and when _n_ = 41, two factors are (83 . 1,231).