Essays/Josephus Problem
Being involved in the development of a programming language has its rewards, none more pleasant than receiving gems like the following (EEM is Eugene E. McDonnell):
No. 6122659 filed 2.13.00 Sun 5 Apr 1992 From EEM To KEI RHUI Subj Josephus Problem With n people numbered 1 to n in a circle, every second one is eliminated until only one survives. For example, for n=10, the elimination order is 2 4 6 8 10 3 7 1 9, so 5 survives. The problem: Determine the survivor's number J(n). J=.1&|.&.#: Try J"0 i.50 for a nice pattern to emerge. See also Graham et al, Concrete Mathematics, Section 1.3.
J can be derived by observing that:
] y=.2^?10$10 2 128 16 32 4 1 64 64 512 8 J"0 y 1 1 1 1 1 1 1 1 1 1 ] y=.>:?10$1000 520 831 35 54 530 672 8 384 67 418 (J"0 >:y) - J"0 y 2 2 2 2 2 2 2 2 2 2
That is, J 2^y is 1 , and (J 1+y)-J y is 2 if 1+y is not a power of 2. As McDonnell asserts, the pattern is made evident by applying J to the first few integers:
>: i.5 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 J"0 >:i.5 10 1 1 3 1 3 5 7 1 3 5 7 9 11 13 15 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37
The statement of J in J is interesting in its own right. 1&|.&.#: is in the form f&.g , defined as follows:
f &.g y g^:_1 f g y
1&|.&.#: y #. 1&|. #: y
That is, convert to the binary representation (#:), rotate by one (1&|.), and invert by computing the binary value (#.).
As a matter of historical interest, binary representation and binary value (denoted by the monads ⊤ and ⊥) were once defined in APL (see Falkoff, Iverson, and Sussenguth [1964]), but were later removed due to space limitations. Such draconian measures are understandable: at the time, APL was running on a S/360 Model 50 with 256 Kbytes of main storage (nevertheless supporting 24 users simultaneously with sub-second response).
References
Graham, R.L., D.E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, Section 1.3.
Falkoff, A.D., K.E. Iverson, and E.H. Sussenguth, A Formal Description of System/360, IBM Systems Journal, Volume 3, Number 3, 1964, Table 1, page 200-201.
Contributed by Roger Hui. Substantially the same text previously appeared in Vector, Volume 9, Number 2, October 1992.