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# Loops

When you need to do a repetitive task, like working through all elements in a list to find a prime or searching a map until you've found all of the gold, loops are a go-to tool.

# Loops: Level 4 Challenges

There are p knights sitting at positions $$[1, 2, ... p]$$ 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 $$C(p, r)$$ 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 = $$C(999,11) + C(121, 12) + C(2, 1) + C(15, 16) + C(99, 3)$$. What is T?

Details and assumptions

$$C(4, 2) = 1$$. 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 $$F(0)=0$$, $$F(1)=1$$ and for $$n \ge 2$$, $$F(n)=F(n-1)+F(n-2)$$.

So, here's my question:

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

###### 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 $$L$$ be the size of the longest possible length of the pattern that can be drawn on an android phone with a $$3 \times 3$$ pattern lock. If the grid is spaced equally by one unit what is $$L \sqrt{31}$$ 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 $$2 \times 2$$ grid, $$L$$ would be $$1 + 2 \sqrt{2}$$.

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