Saturday, August 11th, 2012
Problem 59:

The encryption key consists of three lower case characters. Using cipher1.txt, a file containing the encrypted ASCII codes, and the knowledge that the plain text must contain common English words, decrypt the message and find the sum of the ASCII values in the original text.

Note that the key is repeated cyclically throughout the message utilizing XOR.

Friday, August 10th, 2012
Problem 58:

Starting with 1 and spiralling anticlockwise in the following way, a square spiral with side length 7 is formed.

**37** 36 35 34 33 32 **31**
38 **17** 16 15 14 **13** 30
39 18 **5** 4 **3** 12 29
40 19 6 1 2 11 28
41 20 **7** 8 9 10 27
42 **21** 22 23 24 25 26
**43** 44 45 46 47 48 49

Thursday, August 9th, 2012
Problem 57:

It is possible to show that the square root of two can be expressed as an infinite continued fraction.

sqrt(2) = 1 + 1/(2 + 1/(2 + 1/(2 + ... ))) = 1.414213...

Thursday, August 9th, 2012
Problem 56: Considering natural numbers of the form, ab, where a, b < 100, what is the maximum digital sum?
Digital sum, ie: 47 = 4+7=11

Thursday, August 9th, 2012
Problem 55:

A Lychrel number is a natural number which cannot form a palindrome through the iterative process of repeatedly reversing its base 10 digits and adding the resulting numbers.

Assume that after 50 steps of running the above iterative process, the number is deemed Lychrel…

How many Lychrel numbers are there below ten-thousand?

Thursday, August 9th, 2012
Problem 54:

The file, poker.txt, contains one-thousand random hands dealt to two players. Each line of the file contains ten cards (separated by a single space): the first five are Player 1’s cards and the last five are Player 2’s cards. You can assume that all hands are valid (no invalid characters or repeated cards), each player’s hand is in no specific order, and in each hand there is a clear winner.

How many hands does Player 1 win?

Wednesday, August 8th, 2012
Problem 53: How many, not necessarily distinct, values of nCr, for 1 <= n <= 100, are greater than one-million?

Wednesday, August 8th, 2012
Problem 52:

It can be seen that the number, 125874, and its double, 251748, contain exactly the same digits, but in a different order.

Find the smallest positive integer, x, such that 2x, 3x, 4x, 5x, and 6x, contain the same digits.

Wednesday, August 8th, 2012
Problem 51:

By replacing the 1st digit of *3, it turns out that six of the nine possible values: 13, 23, 43, 53, 73, and 83, are all prime.

By replacing the 3rd and 4th digits of 56**3 with the same digit, this 5-digit number is the first example having seven primes among the ten generated numbers, yielding the family: 56003, 56113, 56333, 56443, 56663, 56773, and 56993. Consequently 56003, being the first member of this family, is the smallest prime with this property.

Find the smallest prime which, by replacing part of the number (not necessarily adjacent digits) with the same digit, is part of an eight prime value family.

Tuesday, August 7th, 2012
Problem 50: Which prime, below one-million, can be written as the sum of the most consecutive primes?