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Alphamatic (aka cryptarithmetic) is a game where digits have been replaced by letters in an arithmetic operation - each letter represents a unique digit, with no two letters having the same value.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. 

Sum of Powers of Numbers - A Derivation of Bernoulli/Faulhaber Formula

Sum of Powers of Natural Numbers - Exciting to Explore Different Ways of Doing It

'While there is one correct answer, there can be multiple ways to solve a problem, encouraging creativity and different perspectives' 

I have always found numbers fascinating - they are versatile, form patterns, help develop creative thinking and more. Satisfaction of solving a problem is a wonderful feeling, greatly enhanced by finding yet another way to do so. 

I remember that one of the first problems at school was summing the first N natural numbers -  the answer S = N(N+1)/2 was amazing; you can set N = 100 and instantly get the sum of all the numbers from 1 to 100.  

In this blog, I wish to discuss several different ways of summing the powers of natural numbers.  The motivation to write this blog came from the way many people reacted to my previous blog about the amazing properties of the number 2025 - in particular the fact that the square of the sum of numbers from 1 to 9 is equal to the sum of their cubes. 

      2025 = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 +9)^2 

        2025 = 1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3 

I can fully understand the confused looks as the result is so counterintuitive, yet true.  Of course, Wiki has a way to make it look obvious as shown in the next slide:  

The figure shows the result for N = 5, but it is easy to peal off squares or add more to demonstrate it for other value of N.  

Sum of Natural Numbers:   First, I shall discuss the sum of first power of natural numbers. We can write the sum in two different ways:

S = 1 + 2 + 3 +...................+ (N -1) + N  ...    eq.1   (N terms here)

S = N + (N -1) + ..................+ 2 + 1       ... eq. 2   (N terms)

Adding the two equations will give us

2S = (N+1) + (N+1) + ..........(N+1) + (N+1)    (N terms)

Or     2S = N(N+1)   giving us   S = N(N+1)/2     ... eq.3

I find it fascinating how maths works:  Multiply eq.1 by 2 to obtain

2S = 2 + 4 + 6 + ...................+ 2N = N(N+1)    ... eq.4  (N terms)

Subtract 1 from each digit in the above equation

      1 + 3 + 5 + ................+(2N -1) = N(N+1) - N    ... eq.5

The LHS has N terms - that is why we subtract N on the RHS

Hence,    1 + 3 + 5 + ....  +(2N -1) = N^2 + N - N = N^2     ...  eq.6

What eq.6 says is that the sum of the first N odd numbers is equal to N^2.   

Eq.4 tells us that  the sum of the first N even numbers is N(N+1) = N^2+N.

I shall now discuss various ways by which we can show that the square of the sum of numbers 1 to N is equal to the sum of cubes of numbers from 1 to N.  We shall also look at the sum of higher powers of natural numbers.

Final Word: We have discussed several methods of summing powers of natural numbers.  The methods have great heuristic value and may be used in a classroom situation to encourage students.

However, the best situation is if the sum of powers may be expressed as a closed expression for all powers - it would then be a matter of working out the expression for a chosen power quotient.  Such an expression is available and is called the Faulhaber's formula.  The derivation of the Faulhaber's formula generally requires advanced knowledge of mathematics.  In Part 2, I shall use a generating function approach to derive the Faulhaber's formula - a method that is tractable with higher school level mathematics.  

The New Year 2025: 2025 is an Amazing Number

My learned friend Prab Bhatt had talked to me about some interesting properties of the number 2025 - the new year that has just started. I share these with some additions:

2025 is a perfect square:  2025 = 45 x 45

In our times, years that are perfect squares happen about once a century.  The previous square year was 1936, and the next will be 2116 - 91 years later.  

Curiously, 2025 is made up of 20 and 25 which add to give 45 - square root of 2025!

2025 is the sum of three squares:  we can write 2025 as follows

                    2025 =  5² + 20² + 40² 

2025 is the product of two squares:  we can write 2025 as follows

                    2025 = 9² x 5² = 3² x 15²

2025 is the sum of cubes of all the digits from 1 to 9:  we can write 2025 as follows

The next year that is the sum of cubes of digits from 1 to 10 will be 3025 - 1000 years later.  Previously, 1296 was the year that was the sum of cubes of digits from 1 to 8!

2025 is the square of the sum of digits from 1 to 9

2025 = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 +9)²

This makes 2025 square of a triangular number, namely 45.  

While 2025 is not a triangular number itself, it is the sum of the 44th and 45th triangular numbers (990 and 1035).

2025 is interesting:  The sum of digits of 2025 is equal to 9.  

                               2 + 0 + 2 + 5 = 9

We can divide 2025 by 3 and the sum of digits of the quotient is again equal to 9. 

2025  ÷ 3 = 675.  Sum of digits of 675 is 6 + 7 + 5 = 18 

                              and the sum of digits in 18 is 9

And again:  675  ÷ 3 = 225.  Sum of digits of 225 is 2 + 2 + 5 = 9 

2025 and number 7:  If we break number 7 in its odd and even parts we have 

                       7 = 6 +1 = 4 + 3  = 2 + 5.  

The product of odd parts raised to the corresponding even part gives

Happy New Year!