Loops: Level 4 Challenges


There are p knights sitting at positions [1,2,...p][1, 2, ... p] at King Arthur's round table. In order to choose the leader, Arthur starts counting at position 1 and dismisses every rth seated knight until only 1 knight remains. The remaining seated knight becomes the leader.

Let C(p,r)C(p, r) be the integer position of the eventual leader when the table starts out with p knights and every rth seated one is dismissed.

Let T = C(999,11)+C(121,12)+C(2,1)+C(15,16)+C(99,3)C(999,11) + C(121, 12) + C(2, 1) + C(15, 16) + C(99, 3). What is T?

Details and assumptions

C(4,2)=1C(4, 2) = 1. The knights start seated at positions 1, 2, 3, 4. Arthur dismisses them in this order: 2, 4, 3. The knight seated at position 1 becomes the leader.

I guess you all familiar with the Fibonacci sequence.

For those who haven't known this term yet, the Fibonacci sequence is a sequence of numbers which F(0)=0F(0)=0, F(1)=1F(1)=1 and for n2n \ge 2, F(n)=F(n1)+F(n2)F(n)=F(n-1)+F(n-2).

So, here's my question:

If F(x)F(x) is the largest prime Fibonacci number where x200x \le 200, find logxF(x)\left\lfloor \log _{ x }{ F(x) } \right\rfloor .

This problem belongs to this set

Android's lock screen feature allows people to protect their phones by joining points on a grid to make a pattern. Let LL be the size of the longest possible length of the pattern that can be drawn on an android phone with a 3×33 \times 3 pattern lock. If the grid is spaced equally by one unit what is L31L \sqrt{31} rounded to the nearest integer?

The rules for drawing a pattern are:

  • You can only use a point once

  • You cannot "jump" a point if it is in between the two points and is part of the line that satisfies the first two points. Ie You cannot go from 1 to 9 without also hitting 5. But you can go directly from 2 to 9 without including any other point.

  • ONCE a point is "taken", it may THEN be skipped. A point cannot be skipped "in transit" only if it hasn't already been taken.

To fully understand the rules, it is best to try them out on a real phone.

Details and Assumptions

  • The points are equally spaced vertically and horizontally by one unit.

  • If it had been a 2×22 \times 2 grid, LL would be 1+22 1 + 2 \sqrt{2} .

Image Credits - stackoverflow, banadia.de

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