http://projecteuler.net/problem=28

Number spiral diagonals

Starting with the number 1 and moving to the right in a clockwise direction a 5 by 5 spiral is formed as follows:

21 22 23 24 25
20  7  8  9 10
19  6  1  2 11
18  5  4  3 12
17 16 15 14 13

It can be verified that the sum of the numbers on the diagonals is 101.

What is the sum of the numbers on the diagonals in a 1001 by 1001 spiral formed in the same way?

http://projecteuler.net/problem=40

Champernowne’s constant

An irrational decimal fraction is created by concatenating the positive integers:

0.123456789101112131415161718192021…

If d_n represents the n^th digit of the fractional part, find the value of the following expression.

d₁  d₁₀  d₁₀₀  d₁₀₀₀  d₁₀₀₀₀  d₁₀₀₀₀₀  d_1000000

http://projecteuler.net/problem=48

Self Powers

The series, 1¹ + 2² + 3³ + … + 10¹⁰ = 10405071317.

Find the last ten digits of the series, 1¹ + 2² + 3³ + … + 1000¹⁰⁰⁰.

Recently, I’ve been typing more Ruby then usual. In Ruby you can have a fairly big number.
http://projecteuler.net/problem=25

1000-digit Fibonacci number

The Fibonacci sequence is defined by the recurrence relation:

F_n = F_n1 + F_n2, where F₁ = 1 and F₂ = 1.

The 12th term, F12, is the first term to contain three digits.

What is the first term in the Fibonacci sequence to contain 1000 digits?