Laws of Nature, Common Sense and Religion in Early Societies (Part 2)

 This is a reposting of a previous feature from 2013.

In Part 1 we had discussed why common sense is fundamentally incapable of explaining and making sense of the world we live in.  Simply, it is a matter of scale.  The world operates at the atomic/molecular level (~0.000001 mm or a nano meter) where particles move near the speed of light (300,000 km/sec) while common sense is built up over centuries from observations made by our senses operating at distances typically greater than 0.1 mm and speeds less than 0.1 km/sec.  Additionally, time scales for understanding fundamental chemical and biological processes are less than a billionth of a second, more than a million times faster than our reaction time.

To understand the working of the natural world using common sense is like trying to understand the geography of the UK by standing at a corner in George Square in Glasgow! An impossible task.

We had built up a whole series of laws by extrapolating what we could observe with our senses - that is until about 500 years ago when technological advances started to extend the limits of our observations. The rest is history; now in the 21st Century, we are truly appreciating the intricacies of the natural world and I believe, correctly formulating the laws governing it - the laws of nature. Theories of Relativity and Quantum Mechanics have made possible the development of new technologies - digital, nano-, bio-, medicine and many others.  Life without these is unimaginable. To somebody living in the 19th Century, the technology today will be science fiction.

Humans like to make sense of things.  With extremely limited empirical information, acquired through our senses, for our ancestors it would have been impossible to explain much of what was happening around us. Natural phenomena like rain, thunder, lightening, tides, storms, earthquakes, motion of heavenly bodies, infectious/mental and other diseases etc. are mediated by causes well outside the range of human perception.  Theories and explanations would be put forward – of course different in different societies – to make sense of such events.  This can give rise to beliefs, customs, rituals, superstitions etc.
Who controls the clockwork-like motion of heavenly bodies; what causes earthquakes and storms? There has to be some almighty, omniscient being looking over the working of nature, controlling everything happening on the Earth.   
Reproduction and death were mysteries.  What happens after death would be a great puzzle.  Losing a loved one for ever is difficult and it would be natural to assume that he/she comes back to life again at some other place and time. Alternately, after death the person would go to some other unknown world where he/she will have access to all the comforts and good life, may be not available when alive.  
A mind full of a large number of unanswerable questions is very receptive to any suggestion that can even partially reduce the bewilderment.  

In distant past humans, with nomadic lifestyle, did not have a great deal of interaction with others.  They were preoccupied with the problems of survival - finding food and shelter was more important than worrying about the philosophical questions of how and why of things.  As humans started to settle in communities - particularly after discovering agriculture - they started to face a whole set of new problems.  In a community, one has to live in close proximity with the neighbours, there is more time to reflect on the natural phenomena around you and people would be living longer so there would be more continuity between generations.  Some sort of tradition will begun to be defined.  

Having evolved from the apes and with the history of struggle to survive, human nature would have the tendency to be selfish - hoarding of food and providing comfort to the family must have been of utmost importance. Physical strength would help to grab more land for growing food and to ensure security.  How does a community survive when everybody is fighting for a bigger share of whatever is available?  There is no point if the mightiest kills all the neighbours; then we are back to nomadic lifestyle. One needs a system of government and rules of conduct and, of course, punishment.      

If I am an intelligent person then I could exploit the situation by solving all the problems in one stroke.  I would attribute all the natural phenomena to an almighty who is benevolent by giving us rain for growing food, wood for making our huts, cows for milk etc. but will punish our collective bad behaviour by bringing storms, earthquakes, diseases and other natural calamities.  Someone who will nourish us but also punish us. He will also tell us how to behave in the community.  This code of conduct will be given to the community through a medium - somebody who can communicate with the almighty being.  Humans are too ignorant and too frail to question what the almighty decides - they must accept without question what is being decided for them.  They must have faith - blind faith is better as this makes the implementation of rules easier ensuring stability of the society. 
We have religion with an almighty God who we must try our best to please.  We have to have a prophet who can bring the message and the priests who can interpret the word of God for the common man.

Let us face it, most of the humans are not that clever.  They are already confused with what is happening around them in nature and about their safety and well-being.  A good lazy way for them will be to follow the code of conduct - it comes with the additional promise of good life after death. To keep reminding the community about the rules of conduct, we have to have beliefs, rituals and superstitions  - if faith is strong then all these make good sense - the main thing is not to question, just follow.

Of course, different regions will have their own unique way of describing God and the rules of conduct, rituals etc.  It does not really matter as nobody is there to question?  If one falls out of line - faith is shaken - then the community can take care of that by removing the unfaithful altogether. The good of the community is paramount and the chief priest can always decide what God wants - he has a direct line to Him.
One can have one God or several Gods looking after different issues. You can have unwritten code of conduct, everything is verbal (easier to modify) or you can have one book or several describing the will of God.  One does not even have to have a God - it could be just an energy field or an abstract being - who is somehow passionately concerned about the earthlings and their welfare.  As long there is unquestioning faith, the system can and has worked.  And it has worked for several millenia.

The question is how do the laws of nature (derived from the empirical evidence) and religion (needed historically to make sense of things) live side by side.  Things will be straightforward but for the idiosyncrasies of the human mind.  We shall leave this for the next instalment.

Laws of Nature and Common Sense (Part 1)

 This is a reposting of a previous feature from 2013

Common sense is what we humans live by – it is kind of wisdom gained from experience over many generations. Humans experience the world through the five senses, but importantly we also use our intelligence to perform cause and effect analysis of what has been happening around us to understand how the world behaves. 
Common sense has worked very well and has ensured the survival of the human race; particularly, in the avoidance of dangerous and harmful situations.  Human societies evolved to preserve and enhance the species.  Shared knowledge and effort proved useful in meeting adverse situations and quality of life improved rapidly as humans started to live together in social groups.  
Survival of human societies is a multifaceted situation; they not only have to guard against the hostile physical environment around them but also against other fellow humans who are driven by greed, violence and other undesirable traits.  Humans have always enjoyed exercising power and control over others around them. This is a classic example of conflicting requirements – living in societies is beneficial but at the same time exposes one to potentially dangerous elements of oppression and exploitation.  How humans dealt with this is fascinating and will be discussed in a future blog.
How does our common sense relate to the laws of nature is what we wish to look at first.  The difference is in the scope of the evidence available.

Laws of nature are deduced from experimental observations at all possible levels of space and time.  As human ability improves to expand such observations to wider regions of space and time, laws of nature are modified or even replaced by a different set of laws.  Fine-tuning of the laws is fundamental to their authenticity and acceptance.  The current set is the best available to make sense of what the empirical evidence tells us. 

Common sense is much more restricted in terms of empirical evidence at its disposal.  Without technological aids - and most human experiences have been in such conditions - humans really occupy a small region roughly in the middle of space-time expanse.   Their experiences are limited in size, speed, time, colour, frequency etc.  Quantitatively: We experience

Physical dimensions from ~0.01 mm to a few hundred km. 
Speeds vary from rest to a few tens of km/hour.
Time is restricted to our reaction time of the order of 10 ms to a few hundred years.
Colour is perceived in the narrow wavelength range of 0.4 to 0.8 micron                                       (1 micron = 0.001 mm)
Perception of sound is limited to frequencies less than about 20 kHz.

Animal species do better in colour and sound perception due to the evolutionary need to protect themselves against predators etc.

Humans mainly perceive the world around them through light which travels at the fantastic speed of 300,000 km per second; this gives the feeling of instantaneous communication – things are happening as we see them.

Natural phenomena like rain, thunder, lightening, tides, storms, earthquakes, motion of heavenly bodies, infectious/mental and other diseases etc. were mediated by causes outside the range of human perception.  Theories and explanations were put forward – of course different in different societies – to make sense of such events.  This can give rise to beliefs, customs, rituals, superstitions etc. 
We shall return to this later in Part 2.

Limited scope of human experience, nevertheless, provided some quite sensible and workable theories about the way the world operates – kind of ‘limited’ laws of nature.  In good true scientific spirit a law would get modified when the evidence against it became overwhelming.  Sometimes it would require great sacrifices as changing a law might encumber on the interests/dogma of the powerful in the society. 

Situation started to improve about 500 years ago with the acceptance that experiment based evidence could not be ignored and must be taken into account in formulating so called laws of nature.  Technological advances helped in removing many of the limits to observations – in space, time and other areas.  Common sense inspired laws of nature started to be replaced by the new physics at the turn of the 20^th century and in the short span of 25 years the acceptance of the new laws of relativistic and quantum physics was overwhelming. 

While common sense laws are arrived at through observations, mostly visual, and rely on the behaviour of matter around us at the macroscopic scale of dimensions greater than about 1 micron (0.001 mm), laws of nature (physics) are determined in the way elementary particles, typically of dimensions less than 0.001 micron, interact with each other.  These interactions are of four types:

gravitational (G), 

electromagnetic (EM), 

strong nuclear (SN) and 

weak nuclear (WN).  

G and EM are long range and affect matter at all distances. SN and WN are relevant only at nuclear dimensions with distances of the order of a billionth of a micron or less!   
Chemical and biological properties are determined by EM forces acting between atoms and molecules at distances of the order of 0.001 microns and may not be properly understood through evidence from everyday experience. Old theories based on human experiences have been unable to make good sense of chemistry and life sciences.
Motion of heavenly bodies is governed by gravitational forces but mostly involves distances far greater than our senses can be sensitive to.  That is why there has been so much confusion about this subject in historical accounts.  Newton's laws of gravitational attraction explained the way heavenly bodies are organised in the sky but a proper understanding of the Universe had to wait for the theories of relativity and the development of nuclear sciences. 

The world of atoms and nuclei was latent to our ancestors as the distances involved were unimaginably small and even their existence could not be contemplated. Understanding the true nature of atoms and nuclei only started at the turn of the 20^th century and this has created the atmosphere in which modern industry could start.  Nuclear energy, Nanotechnology, Computers, Biotechnology, Space Exploration all owe their success to the laws of nature enunciated through empirical evidence acquired through development in technology in the 19th and 20^th centuries.  Without this mind-set, modern industry would not even be science fiction.

Another interesting aspect of this discussion is the emergence of paradoxes.  A language that is weakly developed will be unable to describe a situation that is highly complex – the vocabulary is just not there.  A good example of a paradox is the wave-particle duality in classical physics (mostly based on common sense observations).  In our daily lives we see objects behave like material particles or as undulating waves in motion.  We do not have objects that switch between being waves and particles at different times.  This was the situation until the end of the 19^th century when light was observed to exhibit properties akin to waves in some experiments but behaved exactly like a stream of particles in others.  The situation was resolved and explained by quantum description of light in 1925.  Now, we know that everything in the universe shows wave-particle duality and it is just our attempt to classify things as either waves or particles that is flawed.  We are using the wrong language for our description.

It is not that there are no paradoxes in the modern theories.  There are many and this simply points to the fact that, while we have a much better understanding of the world we live in, we have more to learn and the laws of nature will be rewritten differently in the future.

Introducing Serendipity's Cousins - Leucippity/Aristopity - Validation/Rejection of a Pre-existing Significant Theoretical Insight

This feature is inspired by Professor Sheldon L. Glashow who coined the term Leucippity to describe a hypothesis/prediction that precedes the evidence required for its acceptance as a proper scientific theory.  A classic example of leucippity is Albert Einstein's 1915 prediction of gravitational waves which waited a full century before being experimentally observed in 2015.

In contrast to serendipity which refers to the occurrence of surprising & valuable discoveries that were not actually sought; leucippity refers to the deliberate, often slow, pursuit of experimental validation of an existing theoretical insight. 

The term leucippity is coined in honour of the ancient philosopher Leucippus (5th century BCE) who with his disciple Democritus is credited for founding Greek Atomism that all matter is composed of small indivisible particles which they called 'atoms'.  It took over two thousand years for their idea to become universally accepted!

Examples of leucippity are found in all fields of scientific endeavour - in physical sciences, medicine, geology, mathematics etc. Glashow (1, 2) gives some outstanding examples where the validation of a theory only came after a significant delay.  

The Slide lists a selection of leucippitous discoveries in physical sciences that were awarded a Nobel Prize:   

Leucippity mostly comes in play when a paradigm shift happens in a field of study; new ideas replace old explanations used to make sense of empirical evidence, and in turn generate a host of new predictions.  Such predictions await further empirical confirmation before the new ideas can be accepted as actual real theories (The Scientific Method). Einstein's Theories of Relativity (1905 & 1915) and Quantum Theory (1925) are two classic examples where the new theories explained a host of difficult-to-understand observations of preceding decades but the underlying assumptions in the theories were novel to the extent that few scientists were ready to accept these as real theories.  Einstein was not awarded the 1921 Nobel Prize for his theories of relativity but for his explanation of photoelectric effect - such was the reservations in the scientific community for his 'wild' ideas. Same is true for Quantum Theory.  Thankfully both theories made plentiful predictions and all predictions have been verified experimentally over the past 100 years.  The number of Nobel Prizes awarded for experimental work related to these theories is the proof that these are valid real theories with far-reaching significance for mankind's efforts to understand the laws that govern how nature works - and of course providing very valuable benefits to our every day life.  

We reserve the term leucippity for the verification of ideas/hypotheses that are of significant importance and might have waited for a reasonable time for it. Such ideas are deemed to be important and may substantially influence the progress/development of the field even while the idea is awaiting validation.  If a hypothesis is eventually proved to be invalid then it is likely that it would have done much harm to the scientific progress - particularly if the hypothesis was due to a scientist/philosopher of outstanding reputation. With these considerations, I introduce the word aristopity to describe such a phenomenon.

I have chosen the word aristopity to represent Aristotle (384-322 BCE).  Aristotle was a towering figure in ancient Greek philosophy, known for inventing formal logic and developing a comprehensive system of thought that influenced virtually every field, including ethics, politics, biology, natural philosophy (physics) and arts, essentially laying the groundwork for Western science and philosophy with his emphasis on observation and logical reasoning. 

However, Aristotle did not seek experimental verification of what his theories predicted - his theories were what observations and common sense supported.  The ideas of scientific method and tools (microscopes, telescopes etc.) came much later.  Aristotle's stature was such that his ideas went unchallenged for almost 2000 years and were accepted as eternal truth - to question them could result in getting burned at stake. It is not difficult to appreciate what harm such blind following to his erroneous theories might have done to the development of scientific knowledge. The next two slides list some of Aristotle's scientific ideas that have been discredited.

 

For aristopity, an erroneous hypothesis must survive over a long-enough period of time to affect thinking of a significant number of scientists.  This can be seriously harmful for proper development of scientific ideas. One indeed encounters many aristopitous practices that lasted a long time - for example, astrology, alchemy, phlogiston, Einstein's cosmological constant, Ptolemy's Almagest, Galen's Humourism, Lysenkoism, Phrenology, Miasma Theory and many more.

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Serendipity (Part 1) - Making Unplanned/Unsought Surprising Discoveries - Plays Major Role in All Aspects of Our Lives

 

Serendipity is the faculty of making fortunate and unexpected discoveries by accident                                 ---  Oxford English Dictionary (OED)

Sagacity is the quality of having or showing understanding and the ability to make good judgements.                       --- Cambridge Dictionary

The word serendipity was coined by Horace Walpole in 1754 in a letter to his friend where he recounted a Persian fairy tale 'The Three Princes of Serendip' in which the princes were always making discoveries by accident and sagacity.  Accidents become discoveries because of the sagacity.  In Appendix 1, I reproduce this charming story. 

Serendipity - smooth, pleasing and sweet sounding - has been called one of the most beautiful words of the English language.  Life is full of fortunate accidents and pleasant surprises - one only has to have  awareness/good judgement to own them when they come your way (have sagacity).  Serendipity is the occurrence of surprising & valuable events that were not being sought, often combining chance with sagacity (wisdom/insight).

There is no aspect of life that is untouched by serendipitous happenings - from finding your soulmate at the airport because you missed the flight, or being awarded Nobel Prize by noticing a completely unexpected flicker on a fluorescent screen metres away (discovery of X-rays).

The idea that chance plays an important role in the process of discovery is much older than the use of the word serendipity by Walpole to describe it.  In fact, Robert Hooke in 1679 wrote: " ......it will be much better to embrace the influence of Providence and to be diligent in the inquiry of everything we meet with.  For we shall quickly find that the number of considerable observations and inventions this way collected will a hundred-fold outstrip that are found by design." 

Curiocity, Perception, Cognitive Biases & Dogma:  

Louis Pasteur said 'In the field of observation, chance favours only the prepared mind'.  A chance accidental observation will remain that if the observer does not notice, question and follow through the unexpected finding (exercise perception).  A prepared mind is able to recognise the potential that others might ignore/discard. Curiosity drives the observer to investigate the event/result more in depth resulting in valuable discoveries. 

However, not every chance observation may be converted into valuable discovery nor  everybody is curious and prepared enough to question/investigate the chance offered (not sagacious enough). How can sagacity be encouraged and developed? First, we look into a couple of impediments - our cognitive biases and dogmatic thinking - it is important to enhance cognitive flexibility and actively challenge dogmas (rigid mindset, beliefs based on faith or authority).

We react to what is happening in the world outside according to our perceptions.  These perceptions are interpretation by our brain of the sensory signals mediated by numerous cognitive biases.  The result is that our perception of the world outside almost always represents a 'modified reality'.  

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Perceptions of the Brain - Why Our Perception of Reality is Almost Always Wrong?   A 12-hour course designed for adult education - for non-specialists.

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It is not surprising that in majority of situations, we fail to identify the importance of weak cues because they are overwhelmed by the cognitive biases that we all carry with us (cognitive biases are essential for survival in the complex world, and they are there for a reason).  The role of sagacity is of paramount importance and one needs to resist the  'obvious' interpretation because it is easier and less controversial.  Following Kahneman, in making decisions our brain ignores much of the incoming information to make quick decisions (system 1 thinking) in preference to the more energy intensive process of analysing the signals and then reaching a better/firmer conclusion (system 2 thinking) - this might happen in >90% of the decisions we make everyday. 

Overriding cognitive biases (creating cognitive flexibility) is not easy.  Such biases are deep rooted - I give an example (The Jastrow Illusion) in the following:  

(click on the slide for full page image)

Two identical sized rail-track pieces look very different in size when viewed as in the figure. In my lecture, I asked a student to measure the two tracks with a string - they are exactly the same size. (I encourage you to measure them yourself).  The interesting part is that even though our brain knows that the tracks are identical, it refuses to accept that because of the built-in way it interprets visual signals. The brain continues to ignore the factual information provided.   

It is interesting to explore how one can encourage behaviours leading to serendipity - Reference is a good place to start.   Cognitive biases help us navigate the world safely with ease and efficiently - while not perfect, they have helped human species to survive and indeed flourish. What one needs to enhance cognitive flexibility is a programme of 'improvement' in this sphere.  

One needs to venture out of their comfort zone - learn to accept ambiguities and contradictions that will help the brain to adapt to unexpected situations. 

A healthy brain is most desirable - brain training exercises, solving puzzles, exercise, good nourishing diet will help.

 Mindfulness/meditation improve focus and attention that improves flexible reactions to new situations (helps to be more open-minded).

Learn new skills - a new language for example to improve neuroplasticity.  

Besides improving the internal brain environment, much may be achieved by paying attention to external factors.  Interacting with  people/co-workers from different fields is always helpful in moving away from one's fixed biases and dogmas.   

I end this section by giving an example of how the input from an authority can influence one's decisions:  

End Note:  Many great discoveries have been made where serendipity has played a major role.  Many such discoveries have advanced our knowledge and been crucial to the welfare of the human race.  New research fields like nuclear physics, antibiotics etc owe their emergence to serendipitous discoveries. Indeed, every field has benefitted from serendipitous discoveries and almost a quarter of the Nobel Prizes have been awarded to such accidental discoveries  - and that is only since the year 1900.  Many ground breaking inventions and discoveries since historic times have contributed to the welfare and progress of our species.  In Part 2, I shall list some of the serendipitous discoveries in sciences and other fields of research.  

Of course, serendipity plays a pivotal role in our daily lives and many stories are told about lucky accidents that led to happy conclusions.  

In the end, I wish to point out that many a times, we fail to notice/appreciate the accidental break that is offered to us and we are worse off for that.  A fascinating example relates to Irene and Frederic Joliot-Curie who - not once but two times - failed to own the observations in their experiments with the result that they lost two opportunities of winning a Nobel Prize.  I shall discuss more examples of such 'negative serendipity'  in Part 2.  

Appendix 1:  The story is reproduced from Wiki

"In ancient times there existed in the country of Serendippo, in the Far East, a great and powerful king by the name of Giaffer. He had three sons who were very dear to him. And being a good father and very concerned about their education, he decided that he had to leave them endowed not only with great power, but also with all kinds of virtues of which princes are particularly in need."

The father searches out the best possible tutors. "And to them he entrusted the training of his sons, with the understanding that the best they could do for him was to teach them in such a way that they could be immediately recognized as his very own."

When the tutors are pleased with the excellent progress that the three princes make in the arts and sciences, they report it to the king. He, however, still doubts their training, and summoning each in turn, declares that he will retire to the contemplative life leaving them as king. Each politely declines, affirming the father's superior wisdom and fitness to rule.

The king is pleased, but fearing that his sons' education may have been too sheltered and privileged, feigns anger at them for refusing the throne and sends them away from the land.

The Lost Camel: No sooner do the three princes arrive abroad, they face clues to identify precisely a camel they had never seen.  They conclude that the camel is lame, blind in one eye, missing a tooth, carrying a pregnant woman, and wearing honey on one side and butter on the other.  When they later encounter the merchant who has lost the camel, they report their observations to him.  He accuses them of stealing the camel and takes them to emperor Beramo, where he demands punishment.

Beramo then asks how they are able to give such an accurate description of the camel if they have never seen it. It is clear from the princes' replies that they have used small clues to infer cleverly the nature of the camel.

Grass had been eaten from the side of the road where it was less green, so the princes had inferred that the camel was blind on the other side. Because there were lumps of chewed grass on the road that were the size of a camel's tooth, they inferred they had fallen through the gap left by a missing tooth. The tracks showed the prints of only three feet, the fourth being dragged, indicating that the animal was lame. That butter was carried on one side of the camel and honey on the other was evident because ants had been attracted to melted butter on one side of the road and flies to spilled honey on the other.

As for the woman, one of the princes said: "I guessed that the camel must have carried a woman, because I had noticed that near the tracks where the animal had knelt down the imprint of a foot was visible. Because some urine was nearby, I wet my fingers and as a reaction to its odour I felt a sort of carnal concupiscence, which convinced me that the imprint was of a woman's foot."

"I guessed that the same woman must have been pregnant", said another prince, "because I had noticed nearby handprints which were indicative that the woman, being pregnant, had helped herself up with her hands while urinating."

At this moment, a traveller enters the scene to say that he has just found a missing camel wandering in the desert. Beramo spares the lives of the three princes, lavishes rich rewards on them, and appoints them to be his advisors.

The three princes have many other adventures, where they continue to display their sagacity, stories-within-stories are told, and there is a happy ending.

A Tutorial on Alphamatic Puzzles - Wonderful Brain Training to Enhance Cognitive Function

Since my retirement 20 years ago, I have spent a lot of time in brain training activities - they say that exercising the brain is important for keeping age-related cognitive problems at bay.  I totally subscribe to this theory provided the activities are enjoyable, challenging and address different cognitive domains of the brain.  Playing Scrabble will sharpen a few of the skills relating to vocabulary, spellings, memory, spatial awareness but can only be one of the many different games and puzzles that one needs to indulge in.  I also like alphamatic puzzles as they help analytical reasoning and logic as well as boosting short term memory, attention and concentration - also they require knowledge of only school-level maths and everybody can enjoy them.

Alphamatic (aka cryptarithmetic) is a game where digits have been replaced by letters in an arithmetic operation - each letter represents a unique digit, with no two letters having the same value.  The goal is to find the digits - 0 to 9 - that the letters represent such that the resulting arithmetic operation is true. 

Slides A1 and A2 of the appendix explain the terms used in basic arithmetical operations with particular emphasis on digits carryover.

As a simple example, consider the following puzzle:

In this tutorial, I shall use the notation of numbering the columns from the right (rightmost column is Clm1 etc.) and identify the carry as cif where i is the number of column that generated it and f is the next column to the left of column i,  or f = i + 1) (see appendix slide A1 and A2 for more details) -  for example, the Units column is Clm1 and the digit carried over to the Tens column (Clm2) is c12.  Similarly,  the digit carried over from the Tens column (Clm2) to the Hundreds column (Clm3) is c23.

Also, by convention the number in the leftmost column is not a zero.

Carry plays an important role in solving the puzzle. It is also useful to look at the leftmost column as this can help to give a good idea of the range of values that the letters have in there.  If the sum row contains a higher value column not present in other rows, then the letter in the leftmost column will have a value equal to the carry that may be 1 or 2 (generally the value of a carry is 0, 1 or 2 , and rarely 3) - see slide 1 where A = 1.  If a higher value column is not present then in the leftmost column the sum of all the letters in rows + any carry from the previous column will be <10 - a useful piece of information.

Keeping track of all the details can get quite involved and one may lose direct awareness of the world outside - this might have a wonderful meditative effect that calms the system - particularly if you reach a successful result! 

Let us look at another example to practice the above information:  

Another example of a similar puzzle is as follows:

If the number ABCDEF is multiplied by 3 then it becomes BCDEFA.  Can you find the number?

The solution is given in appendix slide A3 - but try to solve the puzzle yourself first. Interestingly, the puzzle has two solutions showing two different numbers have this property.  Both answers are given in slide A3.

A puzzle that is slightly more difficult is described in slides 4 and 5.  Study the analysis to understand how to approach the solution.  Again, there is no unique way of solving a puzzle and you might wish to try your own method.

Now it is time to try some puzzles yourself!!

Enjoy! 

APPENDIX

 

Simple Templates to build 3X3 Sum and Product Magic Squares

 The history of the sum magic squares (SMS) goes back a long way.  There is something fascinating - almost magical - the way the numbers play out.  

In the following, I shall provide a template for building SMS and also introduce the product (aka multiplicative) magic squares (PMS).  PMS are not well known and few people are familiar with their properties and construction.  The methods described are rather straightforward and can also be enjoyed by those who find mathematics a difficult/confusing subject.  For clarity, I have restricted the discussion to 3X3 magic squares.

First we look at the much better known Sum Magic Squares.

3X3 SUM MAGIC SQUARES:  The SMS consists of 3 rows of numbers, each row having 3 numbers.  The square is an SMS if the sums of numbers in each row, each column, and both main diagonals are the same. 

An example is given in the following:  

                                2     7     6

                                9     5     1

                                4     3     8

The numbers in each row, each column and both diagonals sum to 15, called the magic sum.  What is not generally appreciated is that the sum of middle numbers (shown red below) is equal to the sum of the corner numbers (shown blue below) - it is 20 in this example (4 times the central number).           

                                2     7     6

                                9     5     1                        eq.0

                                4     3     8

Another property of an SMS is that the central number (5 in our example) is always one-third of the sum of rows or columns or diagonals (15 in our example).  Also the sum of all the numbers is 45 that is 9 times the central number (true for all 3X3 SMS)

Template to build an SMS:  A 3X3 SMS has nine elements.  Let us call them a, b, c, d, e, f, g, h and i.  Let the magic sum be equal to S. Then we have:  

                               a    b    c

                               d    e    f

                               g    h    i

where            S =  a + b + c  =  d + e + f  = g + h + i   

or                 3S =  a + b + c + d + e + f + g + h + i            eq.1

We can also write the sum of the two diagonals, the middle row and the middle column as follows

           4S = a + e + i + g + e + c + d + e + f + b + e + h

                = a + b + c + d + e + f + g + h + i + 3e

or       4S = 3S + 3e    (we have used eq.1 here)

Therefore,    S = 3e or e = S/3                                        eq.2

Eq.2 tells us that the centre number is always equal to one third of the the sum S (sum of rows, columns or diagonals).      Therefore, for integral numbers, S must be divisible by 3.

Now we can provide a template for building a 3X3 SMS. 

For this purpose, we chose e = 0, so that S is also equal to zero.  

A template is shown in the following:

                                  a      -a-b      b

                               -a+b      0       a-b                        eq.3

                                  -b     a+b      -a  

The way, I have built the template is by choosing centre number equal to zero and the top corner numbers as a and b.  Since the sum of the numbers in the top row is zero, the top middle number must be -a-b.  The rest follows. 

Now, if we wish to construct an SMS whose rows etc. sum to a number S = 3e then we simply add e to all the elements of the square - to obtain 

                               e+a      e-a-b      e+b

                               e-a+b      e       e+a-b                        eq.4

                                e-b     e+a+b     e-a  

This template also has the property that the sum of middle number of rows and columns is 4e - this is also true for the sum of numbers at the corners.

The SMS in eq.0 is obtained by choosing a = -3 and b = 1. 

3X3 Product Magic Squares (PMS): PMS are set of numbers arranged as a 3X3 square such that the product of numbers in any row, column or diagonal is the same.  An example is given below:

                                18    1    12

                                 4     6     9                          Eq.5

                                 3    36    2

It is easy to check that the product P of the numbers in any row, any column or any diagonal is 216 - this is equal to the central number raised to power 3. The product of the numbers at the middle of outer rows (shown red) and the outer columns (shown red) is 1296 - this is equal to the central number raised to the power 4.

Template to build an PMS: A 3X3 PMS has nine elements.  Let us call them                  a, b, c, d, e, f, g, h and i.  Let the magic product be equal to P. 

Then we have the PMS as:  

                               a    b    c

                               d    e    f

                               g    h    i

where the products of the numbers in the three rows are 

                    P =  a.b.c  =  d.e.f  = g.h.i  

or                P 3 =  a.b.c.d.e.f.g.h.i                     Eq.6

The product of numbers in each diagonal, middle row and middle column is P, therefore

                  P 4 =  a.e.i x c.e.g x d.e.f x b.e.h

or              P 4 =  a.b.c.d.e.f.g.h.i  x e 3  =   P 3 x e 3     

Hence,     P  =  e 3                                       Eq.7

Therefore, the product of the numbers in each row, each column or a diagonal is equal to the cube of the central number. This also implies that for integer numbers a to i in a PMS, the product P must be a whole cube - equal to the cube of the central number e.

Following the procedure for the construction of an SMS  template, we shall first choose the central number e = 1 and two other numbers a and b to give us the following template:  

For a=3 and b=2, eq.9 reproduces the PMS described in eq.5.  We also notice that the product of numbers in the middle of the top and bottom rows and the middle of the 1st and 3rd columns is  P 4 = a4b4  

Similarly, the product of numbers at the four corners is also a4b4      

Hope you have enjoyed this investigation into SMS and PMS.  

Now, it is possible to construct the magic squares so easily.

Have you read:  

  https://ektalks.blogspot.com/2018/10/additive-and-multiplicative-3x3-magic.html         

 

Six Generations Family Tree of the Curie Family

Marie Curie's life holds a particular fascination for me - it signifies a life full of struggle and deep commitment to achieving excellence amidst adversity.  Six Nobel Prizes have been shared within the Curie Family, with Marie Curie being the first and only woman who was awarded two Nobel Prizes in different subjects.  Marie Curie's life is a poignant example of how women were actively and openly discriminated in prevailing male-dominated environment of the early 20th century.   

In such situations, it is natural to ask who her other family members were, what were their achievements and who her descendants are. At home in France, because she was a woman, Marie Curie was denied obvious honours like membership of the French Science Academy. Despite the many roadblocks, Marie Curie not only excelled as a scientist but also won the affection and admiration of the public and the politicians throughout the world.  

In the following, I shall provide Marie Curie's family tree spanning six generations.  I refer you to my blogs (1, 2)  for a detailed discussion of Curie family's scientific achievements. 

For a concise version of Curie Family Tree, please click here.

The organisation of the family tree starts with the central characters (Marie Curie and her immediate family - parents and children).  The following slides then explore the relatives in more detail.  I feel this makes it easier to see the generations without overcrowding.  I acknowledge help from several published accounts of the Curies with a lot of help from the Wiki - these may be reached here (1, 2, 3, 4, 5). 

In the following six slides, each generation is highlighted by a different colour. 

Please click on a slide to see its full page image.  Press Escape key to return to the blog.

A word about the organisation of the slides.  

Slide 1 shows the central characters - Marie and Pierre Curie and their two daughters & their husbands.  The two generations won all six Nobel Prizes - more than any other family in history.

Slide 2 shows Marie Curie's family - her siblings, parents and grandparents.  It is worth mentioning that her sister Dr Bronislawa Dluska was working in Paris and encouraged Marie to come to Paris for studies and provided financial support too to make it possible.

Slide 3 likewise shows Pierre Curie's family - his brother, parent and grandparents.  Pierre and Jacques worked very closely together and did pioneering work in piezoelectricity and magnetic properties of materials.

Slide 4 deals with Eve Curie's husband's family.  Eve Curie had a remarkable career as a diplomat, journalist and author.  She also wrote the first biography of Marie Curie.

Slide 5  describes Irene and Frederic Joliot Curie's children Helen & Pierre Joliot, and grandchildren.  Family tree of Helen Joliot's husband (Michel Langevin) is included because of the historic interest in Marie Curie and Paul Langevin around 1910.

Slide 6  shows Frederic Joliot's family tree with his parent and grandparents researched. 

It has been an exciting experience to update the Curie Family tree.  My talks on the Curies go in detail about their research and personal lives - these may be reached by clicking on the following two links 1  and 2.

The information included in this blog has been teased out of many many websites - the links to all are not easy to list.  I acknowledge all the help I have obtained from Wiki and other genealogy websites.  If anyone wishes to be acknowledged individually then please let me know at ektalks@yahoo.co.uk and I shall be delighted to do so.

Thanks for reading.