A History of APL in 50 Functions

33. Josephus Problem

1∘⌽⍢⊤

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. [99]

The phrase J=.1&|.&.#: in J translated into APL is J←1∘⌽⍢⊤ . It derives by observing that:

   ⊢ y← 2*?10⍴10
2 128 16 32 4 1 64 64 512 8

   J¨ y
1 1 1 1 1 1 1 1 1 1

   ⊢ y← 1+?10⍴1000
520 831 35 54 530 672 8 384 67 418

   (J¨ 1+y) - J¨ 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:

   1+5 10⍴⍳×/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¨ 1+5 10⍴⍳×/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 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 (2∘⊥⍣¯1), rotate by one (1∘⌽), and invert by computing the binary value (2∘⊥).

As a matter of historical interest, binary representation and binary value (denoted by the monads ⊤ and ⊥) were once defined in APL [100], 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).

Appeared in J in [101].