Tim
Varley
Principal Systems Engineer & Technical Lead specializing in distributed systems, AI-driven development, and high-performance infrastructure.
Featured Projects
AI-Native IoT Platform
AI-native IoT platform optimizing water and energy consumption with distributed sensor networks, built entirely using advanced LLM agents.
B2B E-commerce Marketplace
Cloud-native B2B e-commerce platform disrupting the promotional products industry with extreme scalability and 20% cost optimization.
Enterprise SaaS Migration
Transformed legacy monolithic software into cloud-native microservices architecture, achieving 170% performance boost and 99.9% uptime.
High-Frequency Financial Infrastructure
Mission-critical financial data infrastructure delivering embargoed government economic releases to trading systems in under 15 milliseconds.
FinTech Patent Solution
Pioneering multi-channel banking system architecture that earned a US patent and Smithsonian Innovation Award for bridging legacy and digital banking.
Recent Solutions
View All →Champernowne's Constant
An irrational decimal fraction is created by concatenating the positive integers: $$0.12345678910{\color{red}\mathbf 1}112131415161718192021\cdots$$ It can be seen that the $12$<sup>th</sup> digit of the fractional part is $1$. If $d_n$ represents the $n$<sup>th</sup> digit of the fractional part, find the value of the following expression. $$d_1 \times d_{10} \times d_{100} \times d_{1000} \times d_{10000} \times d_{100000} \times d_{1000000}$$
Integer Right Triangles
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 $p \le 1000$, is the number of solutions maximised?
Pandigital Multiples
Take the number $192$ and multiply it by each of $1$, $2$, and $3$: $$\begin{align} 192 \times 1 &= 192\\ 192 \times 2 &= 384\\ 192 \times 3 &= 576 \end{align}$$ By concatenating each product we get the $1$ to $9$ pandigital, $192384576$. We will call $192384576$ the concatenated product of $192$ and $(1,2,3)$. The same can be achieved by starting with $9$ and multiplying by $1$, $2$, $3$, $4$, and $5$, giving the pandigital, $918273645$, which is the concatenated product of $9$ and $(1,2,3,4,5)$. What is the largest $1$ to $9$ pandigital $9$-digit number that can be formed as the concatenated product of an integer with $(1,2, \dots, n)$ where $n \gt 1$?
Truncatable Primes
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. <p class="smaller">NOTE: $2$, $3$, $5$, and $7$ are not considered to be truncatable primes.
Double-base Palindromes
The decimal number, $585 = 1001001001_2$ (binary), is palindromic in both bases. Find the sum of all numbers, less than one million, which are palindromic in base $10$ and base $2$. <p class="smaller">(Please note that the palindromic number, in either base, may not include leading zeros.)
Circular Primes
The number, $197$, is called a circular prime because all rotations of the digits: $197$, $971$, and $719$, are themselves prime. There are thirteen such primes below $100$: $2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79$, and $97$. How many circular primes are there below one million?