There are p knights sitting at positions 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 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 = . What is T?
Details and assumptions
. 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 , and for , .
So, here's my question:
If is the largest prime Fibonacci number where , find .
Android's lock screen feature allows people to protect their phones by joining points on a grid to make a pattern. Let be the size of the longest possible length of the pattern that can be drawn on an android phone with a pattern lock. If the grid is spaced equally by one unit what is 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 grid, would be .