In computer science and mathematics, the Josephus Problem (or Josephus permutation) is a theoretical problem. Following is the problem statement: There are n people standing in a circle waiting to be executed. In each step, a certain number of people are skipped and the next person is executed.

1. The
Josephus
Problem

2. The Josephus Problem
 In computer science and mathematics, the Josephus
problem (or Josephus permutation) is a theoretical problem
related to a certain counting-out game.
 People are standing in a circle waiting to be executed. Counting
begins at a specified point in the circle and proceeds around the
circle in a specified direction. After a specified number of people
are skipped, the next person is executed. The procedure is
repeated with the remaining people, starting with the next
person, going in the same direction and skipping the same
number of people, until only one person remains, and is freed.

3. The Josephus Problem
The problem is named after Flavius Josephus, a Jewish
historian living in the 1st century. According to Josephus'
account of the siege of Yodfat, he and his 40 soldiers were
trapped in a cave by Roman soldiers. They chose suicide
over capture, and settled on a serial method of
committing suicide by drawing lots. Josephus states that
by luck or possibly by the hand of God, he and another
man remained until the end and surrendered to the
Romans rather than killing themselves. This is the story
given in Book 3, Chapter 8, part 7 of Josephus' The Jewish
War

4. The Josephus Problem
Variation of the problem
we start with n people numbered 1 to n
around a circle

5. The Josephus Problem
1
6
7
8
9
10 2
3
4
5

6. The Josephus Problem
1
6
7
8
9
10 2
3
4
5

7. The Josephus Problem
1
6
7
8
9
10 2
3
4
5

8. The Josephus Problem
1
6
7
8
9
10 2
3
4
5

9. The Josephus Problem
1
6
7
8
9
10 2
3
4
5

10. The Josephus Problem
1
6
7
8
9
10 2
3
4
5

11. The Josephus Problem
1
6
7
8
9
10 2
3
4
5

12. The Josephus Problem
1
6
7
8
9
10 2
3
4
5

13. The Josephus Problem
1
6
7
8
9
10 2
3
4
5

14. The Josephus Problem
1
6
7
8
9
10 2
3
4
5

15. The Josephus Problem
1 1
j jn

16. The Josephus Problem
2 1
j jn
1
1jn
j

17. The Josephus Problem
3 3
j jn
1
1
2
1
j
jn

18. The Josephus Problem
4 1
j jn
1
1
2
1
3
3
j
jn

19. The Josephus Problem
5 3
j jn
1
1
2
1
3
3
4
1
j
jn

20. The Josephus Problem
6 5
j jn
1
1
2
1
3
3
4
1
5
3
j
jn

21. The Josephus Problem
7 7
j jn
1
1
2
1
3
3
4
1
5
3
6
5
j
jn

22. The Josephus Problem
8 1
j jn
1
1
2
1
3
3
4
1
5
3
6
5
7
7
j
jn

23. The Josephus Problem
9 3
j jn
1
1
2
1
3
3
4
1
5
3
6
5
7
7
8
1
j
jn

24. The Josephus Problem
10 5
j jn
1
1
2
1
3
3
4
1
5
3
6
5
7
7
8
1
9
3
j
jn

25. The Josephus Problem
11 7
j jn
1
1
2
1
3
3
4
1
5
3
6
5
7
7
8
1
9
3
10
5
j
jn

26. The Josephus Problem
12 9
j jn
1
1
2
1
3
3
4
1
5
3
6
5
7
7
8
1
9
3
10
5
11
7
j
jn

27. The Josephus Problem
13 11
j jn
1
1
2
1
3
3
4
1
5
3
6
5
7
7
8
1
9
3
10
5
11
7
12
9
j
jn

28. The Josephus Problem
14 13
j jn
1
1
2
1
3
3
4
1
5
3
6
5
7
7
8
1
9
3
10
5
11
7
12
9
13
11
j
jn

29. The Josephus Problem
15 15
j jn
1
1
2
1
3
3
4
1
5
3
6
5
7
7
8
1
9
3
10
5
11
7
12
9
13
11
14
13
j
jn

30. The Josephus Problem
16 1
j jn
1
1
2
1
3
3
4
1
5
3
6
5
7
7
8
1
9
3
10
5
11
7
12
9
13
11
14
13
15
15
16
1
j
jn

31. The Josephus Problem
1
1
2
1
3
3
4
1
5
3
6
5
7
7
8
1
9
3
10
5
11
7
12
9
13
11
14
13
15
15
16
1
j
jn

32. The Josephus Problem
1
1
2
1
3
3
4
1
5
3
6
5
7
7
8
1
9
3
10
5
11
7
12
9
13
11
14
13
15
15
16
1
j
jn

33. The Josephus Problem
1
1
2
1
3
3
4
1
5
3
6
5
7
7
8
1
9
3
10
5
11
7
12
9
13
11
14
13
15
15
16
1
j
jn

34. The Josephus Problem
1
1
2
1
3
3
4
1
5
3
6
5
7
7
8
1
9
3
10
5
11
7
12
9
13
11
14
13
15
15
16
1
j
jn
2
n

35. The Josephus Problem
1
1
2
1
3
3
4
1
5
3
6
5
7
7
8
1
9
3
10
5
11
7
12
9
13
11
14
13
15
15
16
1
j
jn
j
jn
32
1
64
1
128
1
256
1
512
1
1024
1
2048
1
4096
1
8192
1

36. The Josephus Problem
Bit
D-B
1
1
2
0
4
0
8
1
16
0
32
1
64
41
N = 41
B-D 1 1 0 0 1 0 19
32+8+1 = 41 16+2+1 = 19
=
=

37. The Josephus Problem
41 19
j jn

38. The Josephus Problem
Recurrent equation
J (1) = 1;
J (2n) = 2J (n) – 1; for n≥1
J (2n+1) = 2J (n) + 1; for n≥1
EVEN
ODD

39. The Josephus Problem
Recurrent equation
J(41) = 2 J(20)+1 = 2 x 9 + 1 = 19
J(20) = 2 J(10)-1 = 2 x 5 - 1 = 9
J(10) = 2 J(5)-1 = 2 x 3 - 1 = 5
J(5) = 2 J(2)+1 = 2 x 1 + 1 = 3
J(2) = 2 J(1)-1 = 2 x 1 - 1 = 1
J(1) = 1

40. The Josephus Problem
Close Formula
41 = 2 +9 = 2x9+15
= 19

41. The Josephus Problem
Thank
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