## An Unusual Sequence

From the 2003 Tournament of the Towns:

Each term in a sequence of positive integers is obtained from the previous term by adding it to its largest digit. What is the greatest possible number of successive odd terms in such a sequence?

## The Vault: Powers of two

My new book of maths puzzles is on its way! It’s packed full of interesting problems to sink your teeth into. I’ll post an update as the launch approaches.

In the meantime, here is a fantastic problem from the USA:

It is given that $2^{2004}$ is a 604-digit number beginning with a 1. How many of the numbers $2^0, 2^1, 2^2, 2^3, ..., 2^{2003}$ begin with a 4?

## A very long number

From the 2000 University of Waterloo Cayley Contest:

The leftmost digit of an integer of length 2000 digits is 3. In this integer, any two consecutive digits must be divisible by 17 or 23. The 2000th digit may be either ‘a’ or ‘b’. What is the value of a+b?

## The answer in the problem

A PuzzleCritic Original:

Let x be the answer to this problem, where x is a positive integer, and let y be the sum of its digits. Calculate 2x-2y.

## Dodgy decimals

From the 2014 American Invitational Maths Examination:

The repeating decimals 0.ababab… and 0.abcabc… satisfy

0.ababab… + 0.abcabc… = 33/37.

Find the digits a, b and c.

## The Vault: A gem from Russia

Here I present my first entry into the Vault, a collection of puzzles so well crafted, they provide the standard by which all other problems should be measured.

From the 1992 Tournament of the Towns:

Let n be a positive integer. Prove that there is a multiple of n, the sum of whose digits is odd.