Problem 65:

The square root of 2 can be written as an infinite continued fraction.

sqrt(2) = 1+1/(2+(1/(2+1/(2+1/…))))

a bit of my work and then some tidbits from around the interweb…

Sunday, August 19th, 2012

Problem 65:

The square root of 2 can be written as an infinite continued fraction.

sqrt(2) = 1+1/(2+(1/(2+1/(2+1/…))))

Saturday, August 18th, 2012

Problem 64:

The first ten continued fraction representations of (irrational) square roots are:

Saturday, August 18th, 2012

Problem 63:

The 5-digit number, 16807=7^5, is also a fifth power. Similarly, the 9-digit number, 134217728=8^9, is a ninth power.

How many n-digit positive integers exist which are also an nth power?

Saturday, August 18th, 2012

Problem 62:

The cube, 41063625 (345^3), can be permuted to produce two other cubes: 56623104 (384^3) and 66430125 (405^3).

In fact, 41063625 is the smallest cube which has exactly three permutations of its digits which are also cube.

Find the smallest cube for which exactly five permutations of its digits are cube.

Friday, August 17th, 2012

Problem 61:

Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae:

Triangle P3,n=n(n+1)/2 1, 3, 6, 10, 15, ... Square P4,n=n2 1, 4, 9, 16, 25, ... Pentagonal P5,n=n(3n-1)/2 1, 5, 12, 22, 35, ... Hexagonal P6,n=n(2n-1) 1, 6, 15, 28, 45, ... Heptagonal P7,n=n(5n-3)/2 1, 7, 18, 34, 55, ... Octagonal P8,n=n(3n-2) 1, 8, 21, 40, 65, ...

The ordered set of three 4-digit numbers: 8128, 2882, 8281, has three interesting properties.

The set is cyclic, in that the last two digits of each number is the first two digits of the next number (including the last number with the first).

Wednesday, August 15th, 2012

Problem 60:

The primes 3, 7, 109, and 673, are quite remarkable. By taking any two primes and concatenating them in any order the result will always be prime.

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.

3736 35 34 33 3231381716 15 141330 39 1854312 29 40 19 6 1 2 11 28 41 2078 9 10 27 422122 23 24 25 264344 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