You are given the following information, but you may prefer to do some research for yourself.
1 Jan 1990 was a Monday.
Thirty days has September. April, June and November. All the rest have thirty-one, Saving February alone, Which has twenty-eight, rain or shine. And on leap years, twenty-nine.
A leap year occurs on any year evenly divisible by 4, but not on a century unless it is divisible by 400.
How many Sundays fell on the first of the month during the twentieth century (1 Jan 1901 to 31 Dec 2000)?
As there are only 16384 routes, it is possible to solve this problem by trying every route. However, Problem 67, is the same challenge with a triangle containing one-hundred rows; it cannot be solved by brute force, and requires a clever method! ;o)
If the numbers 1 to 5 are written out in words: one, two, three, four, five, then there are 3 + 3 + 5 + 4 + 4 = 19 letters used in total.
If all the numbers from 1 to 1000 (one thousand) inclusive were written out in words, how many letters would be used?
Do not count spaces or hyphens. For example, 342 (three hundred and forty-two) contains 23 letters and 115 (one hundred and fifteen) contains 20 letters. The use of “and” when writing out numbers is in compliance with British usage.
Starting in the top left corner of a 2 x 2 grid, and only being able to move to the right and down, there are exactly 6 routes to the bottom right corner.
Lattice paths
How many such routes are there through a 20 x 20 grid?
The following iterative sequence is defined for the set of positive integers: \[n = \begin{cases} \frac{n}{2},\quad &n\text{ is even} \\ 3n + 1, \quad &n\text{ is odd} \end{cases}\]
Using the rule above and starting with 13, we generate the following sequence \[13 \rightarrow 40 \rightarrow 20 \rightarrow 10 \rightarrow 5 \rightarrow 16 \rightarrow 8 \rightarrow 4 \rightarrow 2 \rightarrow 1\]
It can be seen that this sequence (starting at 13 and finishing at 1) contains 10 terms. Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at 1.
Which starting number, under one million, produces the longest chain?
Once the chain starts, the terms are allowed to go over one million.
The sequence of triangle numbers is generated by adding the natural numbers. So the 7th triangle number would be 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28. The first ten terms would be: \[1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \dots\]
Let us list the factors of the first seven triangular numbers:
1: 1
3: 1, 3
6: 1, 2, 3, 6
10: 1, 2, 5, 10
15: 1, 3, 5, 15
21: 1, 3, 7, 21
28: 1, 2, 4, 7, 14, 28
We can see that 28 is the first triangular number to have over five divisors.
What is the value of the first triangle number to have over five hundred divisors?