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?

Tuesday, August 7th, 2012
Problem 49:

The arithmetic sequence, 1487, 4817, 8147, in which each of the terms increases by 3330, is unusual in two ways: (i) each of the three terms are prime, and, (ii) each of the 4-digit numbers are permutations of one another.

There are no arithmetic sequences made up of three 1-, 2-, or 3-digit primes, exhibiting this property, but there is one other 4-digit increasing sequence.

What 12-digit number do you form by concatenating the three terms in this sequence?

Tuesday, August 7th, 2012
Problem 49: Find the last ten digits of the series, 1^1 + 2^2 + 3^3 + … + 1000^1000.

Tuesday, August 7th, 2012
Problem 47: Find the first four consecutive integers to have four distinct primes factors. What is the first of these numbers?

Tuesday, August 7th, 2012
Problem 46: What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?