Tuesday, August 7th, 2012
Problem 45:

Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:

Triangle Tn=n(n+1)/2 1, 3, 6, 10, 15, ...
Pentagonal Pn=n(3n1)/2 1, 5, 12, 22, 35, ...
Hexagonal Hn=n(2n1) 1, 6, 15, 28, 45, ...

It can be verified that T285 = P165 = H143 = 40755.

Find the next triangle number that is also pentagonal and hexagonal.

Tuesday, August 7th, 2012
Problem 44: Pentagonal numbers are generated by the formula, Pn=n(3*n-1)/2. The first ten pentagonal numbers are:

1, 5, 12, 22, 35, 51, 70, 92, 117, 145, ...

It can be seen that P4 + P7 = 22 + 70 = 92 = P8. However, their difference, 70 – 22 = 48, is not pentagonal.

Find the pair of pentagonal numbers, Pj and Pk, for which their sum and difference is pentagonal and D = |Pk – Pj| is minimised; what is the value of D?

Problem 43: The number, 1406357289, is a 0 to 9 pandigital number because it is made up of each of the digits 0 to 9 in some order, but it also has a rather interesting sub-string divisibility property.

Let d1 be the 1st digit, d2 be the 2nd digit, and so on. In this way, we note the following:

Problem 42: The nth term of the sequence of triangle numbers is given by, tn = ½n(n+1);

Using this word list, convert each letter in a word to a number corresponding to its alphabetical position and adding these values we form a word value. For example, the word value for SKY is 19 + 11 + 25 = 55 = t10. If the word value is a triangle number then we shall call the word a triangle word.

How many are triangle words?

Problem 41: We shall say that an n-digit number is pandigital if it makes use of all the digits 1 to n exactly once. For example, 2143 is a 4-digit pandigital and is also prime.

What is the largest n-digit pandigital prime that exists?

Problem 40: An irrational decimal fraction is created by concatenating the positive integers:

`0.123456789101112131415161718192021...`

It can be seen that the 12th digit of the fractional part is 1.

If dn represents the nth digit of the fractional part, find the value of the following expression.

`d1 x d10 x d100 x d1000 x d10000 x d100000 x d1000000`

Problem 39: If p is the perimeter of a right angle triangle with integral length sides, {a,b,c}, there are exactly three solutions for p = 120.

{20,48,52}, {24,45,51}, {30,40,50}

For which value of p 1000, is the number of solutions maximised?

Problem 38: Given the following example:

192 x 1 = 192
192 x 2 = 384
192 x 3 = 576

By concatenating each product we get the 1 to 9 pandigital, 192384576. We will call 192384576 the concatenated product of 192 and (1,2,3)

What is the largest 1 to 9 pandigital 9-digit number that can be formed as the concatenated product of an integer with (1,2, … , n) where n > 1?

Problem 37: The number 3797 has an interesting property. Being prime itself, it is possible to continuously remove digits from left to right, and remain prime at each stage: 3797, 797, 97, and 7. Similarly we can work from right to left: 3797, 379, 37, and 3.

Find the sum of the only eleven primes that are both truncatable from left to right and right to left.

Problem 36: Find the sum of all numbers less than one million, which are palindromic in base 10 and base 2.