Number theory for birthdays and IQ tests

 I have always been interested in numbers and playing with them since I was a small kid. Every time someone has a birthday I am always ready to provide an alternative to have the normal decimal birthday. So e.g. having your 100th birthday when you really have your 49 birthday in decimal numbers.

So here is some number theory for birthdays and IQ tests that you can play around with on your vacations days and prepare for future birthdays and IQ tests. Have fun.

First some short introduction to numbers and the number base. When we use numbers we always assume we're counting with decimal numbers. Decimal numbers means that we are using 10 as the base. Thus when we are saying that someone is 25 years old we really mean that he is 2 * 10^1 + 5 * 10^0 = 2 * 10 + 5 year old. If instead someone has his 25 year birthday in octal number what we are saying is that he has his 2 * 8^1 + 5 * 8^0 = 2 * 8 + 5 = 21 in decimal numbers.

So by varying the number base we can have almost any birthday changed into an even birthday. For example when can we say that we have our 100th birthday. The smallest base is 2, this means that our first 100th birthday happens already at our 4th birthday using base 2. Later in life we can have a 100th birthday when we have 9th birthday, 16th birthday, 25th, 36th, 49th, 64th, 81st and 100th. It is very unlikely that someone will celebrate their 100th birthday in base 11 which would happen at age of 121.

Thus celebrating 100 years happens quite a few times, but not very often still.

Other even numbers are more common. We can have our 20th birthday every second year from our 6th birthday. To be 20, the minimum base is 3 since the number 2 cannot be used in base 2 that only have numbers 0 and 1. Thus 2 * 3 + 0 = 6 is the minimum age to become 20.

However after 6 years old you can always have your 20th birthday at any birthday which is an even number. Thus e.g. at age 38 you will be 20 using base 19, 2 * 19 + 0 = 38.

If you want to search for an appropriate age to celebrate on your next birthday, start by dividing your age into a product of prime numbers. So e.g. 38 is the product of 2 and 19 which both are prime numbers. Thus the most even numbers you can get here is 20 in base 19 and 100110 in base 2. If your age is 18 you have more options, this is divided into prime numbers 2,3 and 3 since 18 = 2 * 3 * 3. So here you can have your 200th birthday in base 3 and 10010 in base 2.

However you stumble into issues with the above approach when the age you have achieved is a prime number itself. So for example when your 37th birthday approaches, how will you divide this into an even number to celebrate. The only obvious even number to reach here is 10 years old which can be achieved with any prime number by using the prime number itself as the base.

Here the age 25 comes to the rescue which is seen as an even birthday by most people. Actually we can prove that every birthday with an odd number of years can become 25 in some base if the odd number is at least 17.

Proof: The proof is fairly simple, first of all an odd number is always written as 2 * k + 1 where k is any number. Second the minimum base to use for an age of 25 is 6 since the number 5 doesn't exist in bases 2,3,4 and 5. Thus the first time to have your 25th birthday happens on your 2 * 6 + 5 = 17th birthday. 

So choose any odd number larger than or equal to 17. This number can always be written as 2 * k + 1 where k is at least 8. But it can also be written as 2 * (k - 2) + (1 + 2 * 2) = 2 * (k - 2) + 5 = 25 in base k - 2. Thus to calculate the number base to use one calculates:

(Odd - 5) / 2. Thus with 37 you get (37 - 5) / 2 = 16. Thus at your 37th birthday you have 25th birthday in base 16.

Isn't it nice to know that you can always claim to be 20 or 25 years old after reaching 17 years of age for the rest of your life :)

Have fun on future birthday in figuring what age you want to have this time.

Actually the base 10 was selected in Arabia, in many older cultures the base 12 was used, even some money systems still have the number 12 in them. If you are working with computer programs it is very popular to use hexadecimal numbers using base 16 with digits 1,2,3,4,5,6,7,8,9,A,B,C,D,E and F.

So on to IQ tests. Most of you have seen tests like the below one:

1, 4, 9, 16, ?

This one is fairly easy, it is the square of the index, thus x^2 is the function in this number series. Thus the next number in the series is 5 * 5 = 25.

Let's take a bit more complex number series now.

2, 9, 28, 65, ?

This one is a bit more difficult to see directly, I will give a hint, it is based on the function x^3 + 1. Thus the next number is 5 * 5 * 5 + 1 = 126.

Now let's take another one, we use the function x^2 - 2 * x - 2.

-3, -2, 1, 6, ?

This looks difficult at the outset, but since we know the answer we cheat and simply set it to 5 * 5 - 2 * 5 - 2 = 13.

So how does one solve this type of IQ tests in a quick manner. Well it is fairly simple using difference techniques, a bit like Fibonaccis tree.

So write the difference between the numbers and then the difference of the differences.

In the above calculation we write it up as follows.

-3,  -2,  1,  6,  13

   1,  3,  5,  7

      2,  2,  2

Interesting the difference is simply a linearly increasing function which is very easy to see and the second difference is simply constant so even easier.

We can see that the difference function is simply 2 * x - 2 and the second difference is simply 2.

For those familiar with derivatives, you can see that 2 * x - 2 is the derivative function of x^2 - 2 * x - 2. and 2 is the second derivative of this function.

So now let's try if this works in practice, here is a number series again:

0, 1, 8, 27, ?

We use the difference technique:

0,  1,  8,  27, => 64

  1,  7,  19, => 37, 

     6,  12, => 18

So we write the answer to be 64. Now let's check the answer, the function I used in this case was:

x^3 - 3 * x^2 + 3 * x - 1

Thus using x = 5 we get 5^3 - 3 * 5^5 + 3 * 5 - 1 = 125 - 75 + 15 - 1 = 64

Thus we found the correct answer of a fairly complex IQ test and we can claim to be more intelligent than we really are :)

Have fun in showing off your capabilities in IQ tests.

Status report RonDB development

 What is going on with RonDB development. Actually a lot, but most happens under the radar at the moment. So this blog will give any interested some idea about what is going on.

RonDB core development is further development of the fork of MySQL NDB Cluster. For the most part this development is focused on our production version RonDB 21.04 that is used at numerous companies in production. Development is very centered around supporting the Hopsworks platform. This means that we now have added 27 new features on top of MySQL NDB Cluster and 127 bug fixes. The latest feature is an improvement of the node recovery. This improvement can bring up to 4-8x shorter restart times. This was seen as an important improvement to ensure that Online Reconfiguration of RonDB in our cloud setting is speedy.

We now have 3 main versions of RonDB core. The RonDB 21.04 that we use in production. RonDB 22.10 that is prepared for use in production. It brings the possibility to store 10x more data in RonDB compared to RonDB 21.04 important for large customers and large applications. We have also started work on the next RonDB generation in RonDB 23.04 that is integrated with MySQL 8.0.33 already.

Managed RonDB has been delivered in two steps. The first integrated the possibility to start up, backup, stop and restore a RonDB database. The configuration is specified in numbers of replicas, number of MySQL Servers and type of VMs for the various node types. One can start the cluster either through a UI or through Terraform.

Now the second step is working as well, this step introduces Online RonDB Reconfiguration. One can change the number of replicas, change the VM types of the nodes and increase/decrease the number of MySQL Servers. This is currently an experimental feature available to our customers on request. The change is fully online and has been verified in internal Hackathons where our developers test various Hopsworks features while the RonDB cluster is reconfiguring.

We are now working on a third step that makes changes more efficient and uses the Kubernetes model with desired state. So the cloud specifies the new desired state and the agent software will ensure that the RonDB cluster moves to this new desired state. Anyone can run RonDB in Docker and try out those new changes on their own laptops.

Those steps are also available using Docker with the rondb-docker github tree. We use Docker as a development platform making it easy to test thousands of state transformations at various levels. Soon there will be videos and blogs describing how to use Docker to test RonDB Reconciliation that will be accessible from the github tree.

It doesn't stop there, a major focus is currently on developing the first version of the RonDB REST API server. This makes it easy to access RonDB using a REST service in parallel with the MySQL Server and more efficient NDB API applications. We have already seen a great interest in this API even before it is completed.

We are also working on automating replication between clusters in different regions.

As usual there is also a set of interesting product ideas on how to improve the RonDB core with even more flexibility in growing and shrinking, making use of SIMD operations to speed up various parts of RonDB and some thoughts on long-term development projects as well.

As usual a benchmark or two is in the works as well. These are further developments of the benchmark described on www.rondb.com where we show throughput and latency of YCSB both in normal operations as well as during recovery.

Laptop vs Desktop for RonDB development

 Most developers today use laptops for all development work. For the last 15 years I have considered desktops and laptops to be very similar in performance and use cases. This is no longer the case as I will discuss in this blog.

Personally I use a mix of laptops and desktops. For me the most important thing as a developer is the screen resolution and the speed of compilation. But I have now found that desktops can be very useful for the test phase of a development project, in particular the later testing phases.

Many years ago I read that one can increase productivity by 30% by using a screen with higher resolution thus fitting more things at the same time on the screen. Personally I have so far found 27" screens to be the best size, larger size means neck pain and smaller means that productivity suffers. The screen resolution should be as high as your budget allows.

My experience is that modern laptops can be quite efficient in compilation. There is very little difference in compilation time towards desktops.

However recently I tested running our new RonDB Docker clusters on laptops and desktops. What I have seen is that the performance of these tests can differ up to 4x.

I think the reason for this large difference is that desktops can sustain high performance for a long time. Some modern desktops can handle CPUs that use more than 200W whereas most laptops will be limited to about 45W. For a compilation that only runs for about 5 minutes and have some serialisation the difference becomes very small. The most important part for compilation is how fast the CPU is on single-threaded performance and that it can scale the compilation to a decent number of CPUs.

However running a development environment for RonDB means running a cluster on a single machine where there are two data node processes, two MySQL server processes and a management process and of course any number of application processes. A laptop can handle this nicely and the performance for a single-threaded application is the same for laptop and desktop. However when scaling the test to many threads the laptop hits a wall whereas the desktop simply continues to scale.

The reason is twofold, the desktop CPUs can have more CPU cores. Most high-end laptops today have around 8-10 CPU cores. The high-end desktops today however goes to around 16-24 CPU cores. In addition the desktop can usually handle more than 4x as much power. The power difference and the core difference delivers a 4x higher throughput in heavy testing.

Thus my conclusion is that laptops are good enough for the development phase together with an external screen. However when you enter the testing phase when you need to run heavy functional tests and load tests on your software a desktop or a workstation will be very useful.

In my tests on a high-end desktop I ran a Sysbench OLTP RW benchmark using the RonDB Docker environment, I managed to run up to 15.000 TPS. This means running 300.000 SQL queries per second towards the MySQL servers and the data nodes. The laptop could handle roughly 25% of this throughput.

Obviously the desktop could be a virtual desktop in the modern development environment. But a physical machine is still a lot more fun.

RonDB is part of the Hopsworks Feature Store platform.

3 commands to start a RonDB cluster

 RonDB is a key-value store with SQL capabilities. We are working on making it really easy to develop applications against RonDB. You can now get a RonDB cluster up and running using 3 commands on your development machine assuming you have Docker installed there.

Here are the commands:

1. git clone https://github.com/logicalclocks/rondb-docker rondb-docker

2. cd rondb-docker

3. ./run.sh

Prerequisites is that you have git installed and Docker or Docker Desktop. Using Docker Desktop and a new Resource Extension one can see the usage of the various containers in both memory and CPU usage. Using it on Windows also requires WSL 2 to be installed.

If you are using Windows it is important that you have set it to use WSL 2 as the engine. One might also have to activate WSL 2 integration with the Linux distribution you are using in the WSL 2. Both of those can be set from the Docker Desktop settings pages. One need to start a new Linux terminal after changing those settings before it actually works.

When trying it on Windows 11 it has worked like a charm for me. But trying it on Windows 10 had issues with firewalls preventing the MySQL Server to start. Feel free to post comments to this blog if you found issues and workarounds for those.

The run.sh command will create the docker image by pulling it from DockerHub. It is a download of a several hundred MBytes, so the time takes depends on the speed of your interconnection. Next it starts a RonDB cluster with 1 MGM server, 2 MySQL Server and 2 Data nodes.

When it started you can access the MySQL Servers on port 15000 and 15001 using a normal MySQL client or the application you are developing.

To access the MySQL Servers you can run the below command using a MySQL client.

mysql --protocol=tcp --user=mysql --host=localhost --port=15000 -p

Enter the password Abc123?e and you are connected to the MySQL Server and can use it as a normal MySQL client connected to a MySQL Cluster. The mysql user have full access to the ycsb% databases, the sbtest% databases, sysbench% databases and the dbt% databases.

You can enter the docker containers in the normal manner using

docker exec -it docker_id /bin/bash

You find the docker_id using the docker ps command.

You can use the run.sh script to create the RonDB cluster of your choice. It has 5 predefined profiles (mini, small, medium, large, xlarge). All profiles have the same nodes except mini which only creates 1 MySQL Server and 1 data node.

We have tested this using Docker Desktop on Mac OS X, Docker Desktop on Windows using WSL 2 and using Docker on Linux. So most developers should be able to try it out in their environment of choice.