CH. I] ARITHMETICAL RECREATIONS 11 



fact that in arithmetic an integral numher is denoted by 

 a succession of digits, where each digit represents the product 

 of that digit and a power of ten, and the number is equal 

 to the sum of these products. For example, 2017 signifies 

 (2 x 10") + (0 x 10 2 ) + (1 x 10) + 7 ; that is, the 2 represents 

 2 thousands, i.e. the product of 2 and 10", the represents 



hundreds, i.e. the product of and 10 s ; the 1 represents 



1 ten, i.e. the product of 1 and 10, and the 7 represents 7 units. 

 Thus every digit has a local value. The application to tricks 

 connected with numbers will be understood readily from three 

 illustrative examples. 



First Example*. A common conjuring trick is to ask a boy 

 among the audience to throw two dice, or to select at random 

 from a box a domino on each half of which is a number. The 

 boy is then told to recollect the two numbers thus obtained, to 

 choose either of them, to multiply it by 5, to add 7 to the 

 result, to double this result, and lastly to add to this the other 

 number. From the number thus obtained, the conjurer sub- 

 tracts 14, and obtains a number of two digits which are the 

 two numbers chosen originally. 



For suppose that the boy selected the numbers a and b. 

 Each of these is less than ten — dice or dominoes ensuring this. 

 The successive operations give (i) 5a; (ii) 5a + 7; (iii) 10a + 14; 

 (iv) 10a +14 + 6. Hence, if 14 is subtracted from the final 

 result, there will be left a number of two digits, and these 

 digits are the numbers selected originally. An analogous trick 

 might be performed in other scales of notation if it was thought 

 necessary to disguise the process further. 



Second Example^. Similarly, if three numbers, say, a, b, c, 

 are chosen, then, if each of them is less than ten, they can be 



* Some similar questions were given by Bachet in problem xn, p. 117 ; by 

 Oughtred or Leake in the Mathematicall Recreations, commonly attributed to 

 the former, London, 1653, problem xxxiv; and by Ozanam, part I, chapter x. 

 Probably the Mathematicall Recreations were compiled by Leake, but as the 

 work is usually catalogued under the name of W. Oughtred, I shall so describe 

 it: it is founded on the similar work by J. Leureohon, otherwise known as 

 H. van Etten, published in 1626. 



t Bachet gave some similar questions in problem xii, p. 117. 



