12/10 



ANSWERS TO THE EXERCISES 



12/10. 1: A possible matrix is : 



12/14. 



12/17. 



12/21. 



13/15. 



13/17. 



12/11. 1: G only. 2: "When at a or b it does not seem to know where it is, 

 and it wanders at random ; c is the only other compartment accessible 

 from a or b; if it arrives in c it seems to recognise where it is, for it then 

 always goes unswervingly through d and e to /, where it stops— perhaps 

 it was always fed there." 



12/12. l:(i) Yes, (ii) I B G 



B 

 G 



J. 



2 



There are no transitions after B. 



2: Yes — for my essential variables! 



1: y must be the identity; ^3 must have no 1 in its main diagonal. 



1: (i) 26; (ii) 52 (see Design for a Brain, S.23/2; here;? = yr). 



1: Two; G's position is completely determined by P's, which has one; 

 y's angle of rotation gives a second. 2: One way would be to move V 

 to mid-way between L and K. 3: One way would be to re-route the 

 air-tube so that it comes down to V instead of up to it. 



1: 3 log2 7, i.e. 8-42 bits. 2: 3 log2 91, i.e. 19-52 bits. 3: 3-3 bits is 

 the minimum, for only 10 combinations are distinct. 4: 1 bit; the number 

 of states and other details are irrelevant. That the answer must be 1 

 bit can be seen by imagining that these are the only two machines 

 possible (as is given), and then imagining that the designer must 

 send his instructions by cable; clearly he need not pay much, for 

 a simple 1-bit distinction is sufficient for the recipient's instruction. 

 5: (i) 49800 bits; (ii) 1-6 bits; no agreement is to be expected, for the 

 values refer not to the one stamp but to two different sets of possibilities. 

 6: n Iog2 n bits. 7: in log2 n bits. 



1: 19 removed. 2: 26 removed. 3: 4-75 bits fell to 300, so 1-75 bits 

 was removed. 4: As ai may go to any of « — 1, and az similarly, the 

 new number of transformations is 



(n - l)(rt - 1) ...(«- 1) (« terms), 

 i.e. (n — 0". Logarithmically the variety was n log2 n and is now n 

 log2 (« — 1), so the variety removed by the restriction is 

 n log2 n — n log2 {n — 1). 

 286 



