ANSWERS TO THE EXERCISES 



2/4. l:No. 2: No. 3: ^, yes; 5, yes; C, no; Z), yes. 4: It must be of the 

 form a->a. 5: Yes; a position with a player mated can have no 

 transform, for no further legal move exists; if C's transformation is 

 closed, every position his move creates can be followed by another, so 

 his transformation can contain no mating moves. 



2/5. 1 : Yes. 2: No; some operands, e.g. 40, end in and will transform to 

 0, which is not in the set of operands. 



2/6. l:n' =n + 10 {n = 1,2,3). 2: a, «' = 7« (m = 1, 2, 3, understoodfor 

 all) ;h,n' = n^;c,n' — l/n;d,n' = 11 — n;e,n' = 1 ; f, «' = n. 

 -,,5 6 7^, . r^ \ ^ 6 7 ,.., ,-101 



3: j 2 3 4^°- "^'-^'^ ^ 25 30 35 ^"^ ^ 2 2 



5: Yes. 6: Yes. 

 2/8. 1: Many-one; both 1 and 8 are changed to 9. 



2/9. 1: No Sale. 2: Maiden over. 



2/10. 1: The main diagonal consists exclusively of I's, and the rest are all 

 zeros. 2: a: ii; b: iii; c: i. 3: a: Yes; b: No. 4: The distributions are 

 the same, the one being merely a reflection of the other. 6:16. 7:4. 



2/11. I: A^: i " ^ 2: The same as the transformation. 3: A. 4: 



' ^ a a c 



n' = « + 2 (« = 1, 2, . . .). 5: n = 49n (n = 1,2,.. .). 

 6: I 



2/14. 1: n" = 9n. 2: a" = a + \6. 3: a'" = 343a. 4: k" = 9k - 4. 



5: m" = log (log m). 6: p" = p^. 7: (i) n = 4« + 9; (ii) n = «4 



+ 2«3 + 2«2 + n; (iii) «' = 1 + 2 log (1 + 2 log n). 8: n' = -27/: 



„ , I + n 2 + n 3 + 2« 5 + 3/? ^ m ^i. -j ^-^ 



— 7 9: « = — ■ — , — ■ , ■. . etc. 10: The identity. 



2 + n 3 +2n 5 + 3n S + 5n, 



12: The limit is at (|, j). 



2/15. 1:2,3,1. 2:g:ll]l 3: A: | ^ ^^ J ^ 4:17. 5:0. 6:9n. 

 7: /. 



2/16. .= V^T: I 2 J ^ j 2: VTU: i ^ ^ ^ <' 



3: They are identical; this equivalence is the chief justification for writing 

 the transformation downwards rather than from left to right. (Cf. 

 Ex. 9/6/8 and 12/8/4.) 



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