Chanukah Candles Mathematics

What is Hanukkah?

Hanukkah is a Jewish festival commemorating the recovery of Jerusalem and subsequent rededication of the Second Temple at the beginning of the Maccabean Revolt against the Seleucid Empire in the 2nd century BCE.

From Wikipedia

How Hanukkah candles are lit?

During Hanukkah festival it’s a custom to light candles on a special type of candelabrum that has eight plus one places for candles. Eight places to light a candle each day and one for an auxiliary candle (shammash in Hebrew) that is intended to light other candles.

During eight days of the holiday a candle is lit in such a way that on first day you light 1 candle, on second day 2 candles and on it goes until on the eighth day 8 candles are lit.

How many are there?

So then the natural question to ask is how many candles do I need to have to be able to light candles for eight days.

The math says sum them up,

(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8) + 8 = …

And you get the number. Remember that additional 8 candles are there since we need to count the auxiliary candle used each day. Even though this calculation isn’t difficult to make, this may quickly change if you need to sum 1 to 100 candles.

What can you do? Is there a quick way to sum the candles? Indeed there is and if you recall your school math you could remember something about arithmetic progression. It gives you the formula below to count any number of candles or any other objects for that matter:

Sn = (a1 + an) * n / 2,

where Sn – sum of n elements, a1 – first element, an – last element, n – total number of elements.

For n elements the formula is

Sn = (1 + n) * n / 2

So in our particular case of Hanukkah it will be

S8 = (1 + 8) * 8 / 2 = 9 * 4 = 36 + 8 = 44 (again don’t forget auxiliary candles)

While this formula is correct one may wonder how do you derive it? What is the intuition? You always can resort to mathematical induction to prove why it works correctly. We leave it for you as an exercise (a free Book Of Proof by Richard Hammack could be helpful). But there is a visual way to understand how this formula was derived.

On the shoulders of giants

It is a folk legend that when one of the greatest mathematicians Carl Friedrich Gauss attended a school a teacher asked pupils to count numbers from 1 to 100 to calm them down. It says that Gauss came up very quickly with the answer 5050.

And the explanation is that he did this trick.

1, … 100 = 101

2, … 99 = 101

3, … 98 = 101

4, … 97 = 101

… …

97, … 4 = 101

98, … 3 = 101

99, … 2 = 101

100, … 1 = 101

He placed numbers form 1 to 100 in such a way that first sequence was ascending from 1 to 100, and the second was descending from 100 to 1. Then if you look at the sum of each pairs it makes 101. There are 100 such pairs, so when you multiply them you get

101 * 100 = 10100

But we counted each pair twice, so there is a need to divide this number by two.

101 * 100 / 2 = 5050

You may think at this point, well Gauss was a genius so he came up with this explanation, but it still feels like I do not get it fully. And I agree. It is still not fully clear how he came up with this trick. So let’s try to look at it more visually.

Visual aids to the rescue

If you look at candles that are lit on each day of Hanukkah from above we’ll see something like on the figure below.

You’ll see that this figure resembles a lot a right triangle. So it’s good to think about this in this way for a moment.

Now, you may recall that to get an area of a rectangular you multiply its two sides

Area = a * b

Well, rectangular is build out of two right triangles isn’t it? So if you look carefully at the figure below you’ll see what Gauss possibly thought.

You have 8 candles on one side of the rectangular. Then you have 9 candles on the other side of the rectangular. The area of this rectangular is

Area = 8 * 9 = 72,

but since we are only interested in half of the rectangular then let’s divide it by two.

72 / 2 = 36.

And don’t forget about adding 8 auxiliary candles.

So we get

36 + 8 = 44

Let there be light

That’s it folks. Let the light of the candles enlighten us in math.

Take care.


Real examples of inventions. Inventions are all around us.

Some definitions

This post is another one in the series of posts about Inventions and Theory of Inventive Problem Solving (abbreviated in Russian as TRIZ). First, I’d like to call it Inventiveness Theory for short and I’ll use it interchangeably with TRIZ. Because I find it very strange that the Russian acronym ТРИЗ was translated into English as TRIZ and used throughout. Also the Theory of Inventive Problem Solving sounds very cryptic and confusing to an English speaker to say the least.

Why do I write these posts?

The main urge to right these posts is to share the excitement I have about how Inventiveness Theory can help ordinary people to fell like they have tools to be creative in ways that they could hardly imagine, since inventions and inventors are covered with the mythology of divine inspirations and thousands of trial and error attempts before being able to come up with an invention. Inventiveness Theory shows that it’s not exactly the case and its tools and methodology can direct you to invention using algorithmic approach.

Inventiveness is in every engineering field

In the previous post I provided examples of how Inventiveness Theory uses standard solutions to problems that have similar structure. In this post I’d like to show inventiveness is used in every human endeavor. For example, it’s difficult to think of any progress in engineering, physics, mathematics, chemistry without engineers and scientist constantly resolving contradictions and this way making inventions. If you recall Inventiveness Theory defines invention as a resolution of a contradiction that the problem presents.

When Genrikh Altshuller and Rafael Shapiro first wrote about TRIZ in 1956 article1 most of the examples they provided for the applications of this approach were from mechanical engineering. Later Altshuller and his students worked on more advanced version of Inventiveness Theory which was applied to electrical engineering, chemistry and other fields. Only in recent decades TRIZ was applied to electronics, software engineering and other disciplines.

Since, I have an experience in software development and software testing I’d like to provide most of my examples from these fields. But, I’ll also provide examples from electronics, physics and mathematics. It turns out there could be no science or engineering as we know it without creativity.

Computer Science


If we look at Computer Science history we’ll see that it evolved from invention to invention. First, there were mechanical calculators, then came electro-mechanical ones using relays, then electronic computers using lamps. Then with an advent of the semiconductors computers were made with hundreds of transistors later to be replaced with microchips having billions of transistors in one single microcircuit.


Software too had an interesting evolution moving from machine language programming to invention of compilers that allowed a more abstract approach to programming to high level programming languages like Java, Python etc. that allowed programmers to reason almost in a human language while righting programs.

Each of these evolutions wouldn’t be possible without a chain of inventions that scientist and engineers made along the way. For example, let’s take Data Structures and Algorithms topics that any programmer learns to some degree. Sorting algorithms like bubble sort or merge sort are an example of inventions when a problem of sorting data is resolved by various creatives ways, like swapping as in bubble sort or divide and conquer as in merge sort approaches.

What is interesting is that in TRIZ there are a number of ways to resolve a physical contradiction2 which are

  • In space
  • In time
  • In structure
  • By condition

And in Computer Science computational complexity is also measured in space by memory storage requirements and in time it takes for an algorithm to run.

Real example

One concrete example of a contradiction in software engineering is how to be able to update software without the need to rebuild it which requires extensive resources and procedures once it is deployed in production. The contradiction here is that software should have certain parameters to function, but there is a need for these parameters to be changed when required. This particular problem was resolved in structure by extracting parameter values into a dedicated configuration file. This file is loaded by the main program on start up and can be even reloaded on the fly. This approach allows to update the parameters as required without a need to invest additional development time and rebuilding the software.


  1. Altshuller, Genrikh, Shapiro, Rafael. “On Psychology of Inventive Creativity”. Questions of Psychology, no. 6, 1956, p. 37-49.
  2. Petrov, Vadimir. TRIZ Basics: Theory of Inventive Problem Solving. Self publishing, Kindle edition 2019, p. 368.

The same trick used all over again. Using standard solutions.

In this post I continue to talk about how the methods from the Theory of Inventive Problem Solving (abbreviated as TRIZ in Russian) can be used by anyone to solve day to day challenges.

This time, again we’ll look at the problem I had in my house and how it was resolved using inventive approach.

Inventive situation

The door lock handle that you see at the top of this post is a typical one that is used across North America, particularly, in Canada. The side of the lock that has a handle resides inside the house. The main issue I had with it was that it was very difficult if not impossible to understand what position handle was in. Is it locked or unlocked, especially in evenings when the lighting conditions are poor. Just look at the image below and tell me whether you can see the handle at all?

As you saw the handle blends with the circular base it attached to and is hardly visible if at all.

As in the previous post where there issue was to find who had stolen a tire valve cap the contradiction is that a cap and in this case a handle has the same color as the base of the lock and is poorly discernable to understand what the sate of the lock is.

You possibly guessed that as in the previous case we possibly can play with the shape of the handle, its color or something else. Again, Ideal Final Result in this inventive situation is for the handle to notify by itself that it’s locked or unlocked. And as you, probably, guessed correctly we can use the same solution as before, namely, we can color it in a white color with wite-out.

And this is what I did at first as can be seen in the image below

This wasn’t a bad idea, but I thought I can do even better, especially at night, since the white color doesn’t glow in the dark. But a luminous tape does. So that was what I did, I used such a tape. When the lights are on it absorbs light and later it emits it and this way the tape glows. This allows you to see at a glance whether the door is locked or unlocked as can be seen below

As you’ve noticed the solution to this problem looks very much like the one before. And actually, in TRIZ there are typical problems and non-typical problems. When you see that a problem at hand is a typical problem it means that there are already existing solutions that you can use out of the box to solve it. In TRIZ one such toolbox is Standard Inventive Solutions also known as Standards. This can be seen in the diagram taken from Vladimir Petrov’s book TRIZ Basics.

Petrov, Vladimir. TRIZ Basics: Theory of Inventive Problem Solving, 2018

Ideal Final Result. Or how to solve it?

What is it all about?

As I mentioned in the previous post I’d like to share with you how anyone can be an inventor and solve everyday problems from simpler to not so using the methodology from the Theory Of Inventive Problem Solving (aka TRIZ in Russian).

Today we will look at a real problem that I faced while living in an apartment building in Israel.

Inventive problem

That apartment building had a public storage where tenants could store their bicycles and other stuff. So it was natural for me to store my own bike there as well. But it happened that someone stole the black valve cap on one of the tires of my bike.

This is how the valve cap I am talking about looks like

How to solve it?

Well, how would you solve this issue? I can tell you right away that at that time there were no cameras installed in the the storage room, and installing one would be prohibitive. It was possible not to do anything as well as a solution.

So let’s try to approach this problem using tools from TRIZ. As I mentioned before that’s first identify a contradiction that happens in this situation. I want to have my valve cap in place without people ‘borrowing’ it. Then more specifically, we may say that we want the cap to be distinguishable if it is stolen, and ideally not to be stolen in the first place. This is what called in TRIZ as administrative contradiction.

The issue lies in the fact that the valve cap is black on any of the bikes and it is indistinguishable when stolen. So the technical contradiction as it’s known in TRIZ, means that the cup isn’t distinguishable when it should be clearly distinguishable.

Ideal Final Result (IFR)

Now, it’s time to introduce one more term, which is an Ideal Final Result. It will help us to focus the contradiction and to resolve it. Ideal Final Result means that we are interested in such a state when the problem that exists resolves by itself, or it does not require any solution at all. When we think about the valve cap issue this way the solution becomes crystal clear.

How about making the valve cap to notify us by itself that it was stolen? One way to achieve it is to have a valve cap that has a different structure, like shape or color. Well, indeed changing the color of the cap was what I did. I took a BIC wite-out and simply colored the cap in white color! Yep. That solved the issue, since even if the cap was stolen I’d see it right away on another bike.

This approach solved the issue and also made the valve easier to spot when I needed to inflate the tire.

Well, thank you for reading until the end and stay tuned for the next blog post in the series.

Take care.

Inventions are everywhere. You just need to look carefully.

From idea to invention by an algorithm

I think I’ve exhausted the nutrition and fasting topics, so I decided to move on to something else like creativity and inventions. You’ve heard it right, inventions. I’ve already posted a number of posts on this subject before. It happened that around 2004 I stumbled upon an article by Pavel Amnuel about how to come up with sci-fi ideas using an algorithm and since then I was hooked. Through Pavel Amnuel I learned about Genrikh Artshuller a soviet sci-fi writer and an inventor of the Theory of Inventive Problem Solving (aka TRIZ in Russian). Later I met with Pavel Amnuel who was a sci-fi writer and astrophysicist himself. Reading articles and books at website I came alone Searing Mind short sci-fi story that Altshuler wrote. So I decided to translate it into Hebrew which I did together with Israela Beker. Through this work I came to know Vladimir Petrov who was an inventor and an engineer and also Altshuller’s close student as Pavel Amnuel was.

Everyone can be an inventor

Since then, I am back and forth with regard to TRIZ having read a number of books that Genrikh Altshuller wrote and trying to use his approach in a real life. So I thought to myself that I could write a series of posts where I’d describe how it was possible to use TRIZ methods in solving every day’s problems. How it is possible to approach various issues that each one of us face in everyday life. As one author wrote all that we see around us that was artificially made was once a one’s person idea. And it’s correct. Anything from houses, furniture, cars, computers didn’t exist and had to be invented. Actually, almost everything that we deal with on a daily basis which is a non-leaving matter was created by humans.

What I intend to write?

So there you have it a post that is dedicated to looking at a problem and how it was solved using TRIZ methodology. By the way what I refer to as a problem is actually defined as a contradiction in TRIZ. Contradiction lies at the heart of TRIZ, since its machinery is useful only when contradiction found and clearly reviewed to be later resolved. By the way, invention defined in TRIZ as a resolution of a contradiction. So if there was no contradiction there was no invention either by TRIZ definition. This makes it much more easier to decern among patents what is actual genuine invention in comparison to being a clone or a modification of an existing invention.

Next post will be such a post as described above. Till then take care.

68 hours fasting results

How did I end up doing it?

Previously I did a 42 hours fast. After that I returned to doing the usual 19 hours intermittent fasting with 5 hours eating window. But I thought to myself that I could do even better and try a longer fast to loss even more body fat as a result. So I tried and finished a longer fast which was scheduled to be 66 hours, but in the end ended up being 68 hours. The fast ended with a big surprise that I’ll discuss later.

Initial setup

Fasting length

From 4:20 PM on Monday, Nov 21th to 12:20 PM on Thursday, Nov 24th.

Fasting progress

Detailed progress

Same information as a graph

Final result

After 63 hours of fasting at 7:20 AM, Nov 24th

A big surprise

The fast has been finished successfully. This time it was more difficult than 42 hours fast. What is funny I did 26 hours more this time, but weight loss was the same as during 42 hours fast.

  • Initial weight was 78.1 kg (172.18 lb), body fat 24.8 %
  • Final weight is 74.4 kg (164.02 lb), body fat 23.4 %