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 \( 1 + 2 \sqrt{2} \).

Image Credits - stackoverflow, banadia.de

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