**The survivor label = (((((0+m1%2)+m2)%3+m3)%4 +m4)%5+m5)%6....**

Examples:

Player : [0..5]

If m=3,

{0,1,2,3,4,5}

=>{0,1,3,4,5} - 2

=>{0,1,3,4} - 5

=>{0,1,4} - 3

=>{0,1} - 4

=>{0} - 1

Saved=(((((0+3)%2+3)%3+3)%4+3)%5+3)%6

= 1->1->0->3->0

Player : [0..4]

If m=3,

{0,1,2,3,4}

==>{0,1,3,4} -2

==>{0,1,3} -4

==>{1,3} - 0

==>{3} - 1

Saved=((((0+3)%2+3)%3+3)%4+3)%5

= 1->1->0->3

Player : [0..3]

If m=3,

{0,1,2,3}

==>{0,1,3} -2

==>{0,3} - 1

==>{0} - 3

Saved=(((0+3)%2+3)%3+3)%4

= 1->1->0

Player : [0..2]

If m=3,

{0,1,2}

==>{0,1} -2

==>{1} - 0

Saved=((0+3)%2+3)%3

= 1->1

Player : [0..1]

If m=3,

{0,1}

==>{0,1} -0

==>{1}

Saved=(0+3)%2

= 1

Player : [0]

No need to play!

Saved= 0

But, Why?

====================

Proof.

If there is n people, last killed guy is X, (X+m)%n will be the next guy to be killed.

If there is n people, killed guy is Y, (Y-m)% (n) will be the prev guy to be killed.

and all label after >= (Y-m)%(n) will be +1, then the last guy labelled as (Y-m)%n;

Ok, we do the process backward.

If the saviour guy is 0. add dummy 1.

(0-m)%2 will be the prev guy to be kill => (0-3)%2 =-3%1=1.

If we enforce prev guy to be zero, all other guys need to advance to position (?+m)%2, ?={0,1}

0->1

Then we add this new prev guy as '0'

1->0

The total displacement of the surviour = 0+m%2 = 1.

Result:{0,1}

Then in next round, added a dummy 2.

(0-m)%3 will be prev guy to be killed = -3%3=0

If we let this guy to be zero, all guys need to advance to position (?+m)%3, ?={0,1,2}

1->1

2->2 (dummy)

Then we add this new guy as '0'.

0->0;

The total displacement of the surviour = ((0+m%2)+m)%3

Result:{0,1,2}

Then in next round, added a dummy 3.

(0-m)%4 will be prev guy to be killed = -3%4=1

If we let this guy to be zero, all guys need to advance to position (?+m)%4, ?={0,1,2,3}

0->3

2->1

3->2 (dummy)

Then we add this new guy as '0'.

1->0;

The total displacement of the surviour = (((0+m%2)+m)%3+m)%4

Result:{0,1,2,3}

Then in next round, added a dummy 4.

(0-m)%5 will be prev guy to be killed = -3%5=2

If we let this guy to be zero, all guys need to advance to position (?+m)%5, ?={0,1,2,3,4}

0->3

1->4

3->1

4->2 (dummy)

Then we add this new guy as '0'.

2->0;

The total displacement of the surviour = ((((0+m%2)+m)%3+m)%4+m)%5

Result:{0,1,2,3,4}

Then in next round, added a dummy 5.

(0-m)%6 will be prev guy to be killed = -3%6=3

If we let this guy to be zero, all guys need to advance to position (?+m)%6, ?={0,1,2,3,4,5}

0->3

1->4

2->5

4->1

5->2(dummy)

Then we add this new guy as '0'.

3->0;

The total displacement of the surviour = (((((0+m%2)+m)%3+m)%4 +m)%5+m)%6

Result:{0,1,2,3,4,5}

* If m is changing in each round, the equation will be:

The total displacement of the surviour = (((((0+m1%2)+m2)%3+m3)%4 +m4)%5+m5)%6.....

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