Sunday, 22 April 2018

What is the Number(s) of the Torn Page/Pages? - Two Seemingly Impossible Puzzles

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I have two seemingly impossible puzzles below:

PUZZLE 1: One page of a book is torn off and the sum of the remaining page numbers is 14001.  Can you find the torn page number?

PUZZLE 2: Some pages were torn off from a  magazine with page numbers from 1 to 32. If the sum of the  remaining page numbers is 495, how many pages were torn off and what are their numbers? 

Puzzle two is more tractable but puzzle one has created a lot of confusion and many wrong and misleading statements and solutions have been posted online.  
I write this blog to provide proper solutions to these two very interesting mathematical puzzles.

Both puzzles appear to provide too little information for solution.  
However, the details are in the way books and magazines are printed and their pages numbered.  Following standard practice, I shall call the leaf of a book a page which has two page-faces (two page numbers); I also note that all page-faces are numbered and start from number 1 on page one of the book.   

The statements of the puzzles may be supplemented by the following observations:

1.  Pages in a book are created by printing four page-faces (2 in the front and 2 in the back) on one sheet of paper and then folding the sheet in half.  Books are stitched and bound in a way that the page number of the end page-face is a multiple of 4
2.  The sum of all page numbers is an even number. (refer to point 1) 
3.  Each page starts with an odd number on the front and has an even number on its reverse side.  
4.  The sum of page numbers on the two page-faces of a page is an odd number (ref. point 3).
5.  All page numbers are integers.

In puzzle 1, one page is torn off the book; Therefore, the sum of remaining numbers must be odd (ref. points 2 and 4)
This has been a problem with several statements of puzzle 1 where the sum of remaining page numbers is given as 10,000 or 15,000 etc - an even number.  This is incorrect and makes the puzzle unsolvable.

SOLUTION (PUZZLE 1):  One page of a book is torn off and the sum of the remaining page numbers is 14001.  Can you find the torn page number?

Let the page numbers in the book go from 1 to N.
Also suppose that the page with page numbers P and P+1 has been torn.
Let us work out the sum S of  page numbers 1 to N in the book.  It is

        S = 1 +    2    +   3 + 4 +   ........+ (N-1) + N
OR    S = N + (N-1) + ....................    +    2    + 1

Add the two equations

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

OR    S =  N (N+1)/2          eq.1

The sum of numbers on the torn page = P + (P+1) = 2P + 1

Therefore, the sum of remaining page numbers, 14001, may be written as

          N (N+1)/2 - (2P + 1)  = 14001                   eq.2

or      N^2 + N = 4P + 2 + 28002 = 4P + 28004      eq.3

We note that the minimum value of P is 1 (first page is torn) and the maximum value of P is N-1 (last page is torn).  
These limits of P, when used in eq.3, will define the range that N may have.

For P = 1 _____  N^2 + N = 28008
                         N^2 + N + 0.25 =  28008.25
                         (N + 0.5)^2 = (167.36)^2
Therefore           N = 167.36 - 0.5 = 166.86

For P = N-1_____  N^2 + N = 4N - 4 + 28004
                            N^2 -3N = 28000
                            N^2 -3N + 2.25 = 28000 + 2.25 = 28002.25
                            (N - 1.5)^2 = (167.34)^2
                            N = 167.34 + 1.5 = 168.84

Therefore, The maximum page number in the book must be in the range 167 to 169.  But N also has to be an even number.  This gives a unique value of N as 168. We can use this value of N in eq.2 to calculate P - the torn page number.

                      14001 = 168 x 169/2 - (2P + 1)
or                     2P = 14196 - 14001 -1 = 194
or                         P = 97

The answer of puzzle 1 is that the page with page numbers 97 and 98 was torn off.  This answer is unique and no other number satisfies the equations.

SOLUTION (PUZZLE 2): Some pages were torn off from a magazine with page numbers from 1 to 32. If the sum of the  remaining page numbers is 495, how many pages were torn off and what are their numbers?

Sum of the page numbers (all pages present) of the magazine = 32 x (32+1)/2 = 528
After tearing off some pages, the sum of the remaining page numbers = 495

Sum of the page numbers torn off = 528 - 495 = 33

The sum of the front and back page numbers for a page with front face page number P is P + (P+1) = 2P+1

If an even number of pages are torn then the sum of their page numbers will also be even - this is inconsistent with the odd number 33.
Therefore, an odd number of pages were torn off. 

The answer cannot be 1 page torn - then the page numbers will have to be 16 and 17 to add to 33.  However, in a magazine the number on the front page face is always odd - not even as in 16.

We can now find actual torn page numbers by trying 3, 5, 7... pages. 

For 3 pages, I calculate the following combinations which add to 33.

Front page face numbers 1, 3 and 11 give (1+2) + (3+4) + (11+12) = 3 + 7 + 23 = 33
Front face numbers 1, 5 and 9 give (1+2) + (5+6) + (9+10) = 3 + 11 + 19 = 33
Front face numbers 3, 5 and 7 give (3+4) + (5+6) + (7+8) = 7 + 11 + 15 = 33

For 5 pages, the lowest sum will be if the first five pages were torn.  This will give us 
(1+2) + (3+4) + (5+6) + (7+8) + (9+10) = 55 which is greater than 33.  Therefore 5 or more pages are not possible answers.

The three torn pages have page numbers:
1,3,11   or  1,5,9   or  3,5,7

Note that page numbers 3,5,7 represent consecutive pages.  If the puzzle is worded as consecutive pages were torn off the magazine then 3,5,7 becomes a unique answer.  
This may be found by the following method.  
The page numbers of the three torn pages are
            P, P+1, P+2, P+3, P+4 and P+5.  
Their sum is 6P+15. 
If this is equal to 33 then P = 3 as before. 

Hope you have enjoyed the puzzles.  Let me know by writing to or by leaving your comment on the blog site.    

Wednesday, 18 April 2018

Division of a Huge Cake: An Interesting Mathematical Puzzle

Mathematical puzzles are addictive and lately, my addiction to them has become quite strong.  The ones I like most are those that have some unexpected outcome and, of course, they also should require some struggle to find the solution.  

Division of a Huge Cake is a puzzle which was published in The Guardian a few years ago.  I thought the published solution of the puzzle did not quite do the justice to the beautiful and somewhat counter-intuitive outcome.  This encouraged me to write this blog - hope you enjoy the discussion of the result.  

The puzzle is (reformulated by me): An eccentric duke throws a big party for 100 of his good friends.  He wishes to give bigger portions of the cake to some of his selected important guests who are served first.  The duke concocts a strange (or shall we say eccentric) formula for dividing the cake  
The first guest gets 1% of the cake.  
The second guest gets 2% of the remaining part.  
The third guest gets 3% of the part remaining and so on...  .  
The 50th guest gets 50% of the part left, and 
the 100th guest gets 100% of whatever is left.
Who gets the biggest piece and how much does he get?  Did the duke achieve what he wanted?
Is this a good way to divide the cake?

Solution:  As the guests are served their share of the cake, the amount remaining gets progressively smaller and even though they are taking a bigger percentage of the remaining cake, after a certain stage the size of the serving starts to get smaller.  One of the guests will get the biggest piece and we need to work it out by using some algebra:

Let us say that for guest number the amount of cake left is a fraction P of the original size.  
Guest N will get N% of this; that is - Guest N receives a portion = p x N/100 
After guest N has had his portion, 
the cake left is equal to   P - P x N/100  = P (1 - N/100)
The next guest (number N+1) will take (N+1)% of this remaining size i.e., 
his share will be  (N+1)/100 x P (1 - N/100)

The question is - for what value of N,  the share that guest number (N+1) receives is smaller than the share taken by guest number N?  
Put it another way, we want to find out for what N, the following relation is true:

(N+1)/100 x P (1 - N/100) - P x N/100 < 0

Multiply by 10,000 and divide by P to get:

(N + 1) x (100 - N) - 100 x N < 0 

OR   N x 100 + 100 - N^2 - N - 100 x N < 0

OR   N^2 + N > 100

Equation in red tells us that when N is large enough for the inequality to hold, subsequent guests will get a smaller portion than the previous guest.

Obviously for N less than or equal to 9, the left hand side is smaller than 100; but for N = 10, the left hand side is 110 and is greater than 100.  
One can use a spreadsheet to work out the portion of the cake that each guest receives.  This is shown in the slide below (click on slide to see full page image)

Guest number 10 will receive the biggest slice of the cake and he will receive 6.28% of the initial cake.

Notice that under this plan, guest 30 onwards receive very little cake - guest 32 only receives 0.1% of the cake. Interestingly, guests number 85 onwards receive less than a molecule of the cake - there is no cake left for them to share.

Is there a more equitable way of dividing the cake?  The problem with the duke's method is that the portions served in the beginning are too large.  Consider a 10 kg cake with an average serving of 100 g per guest.  In duke's method, guest number 10 will receive 628 g portion!  
This situation may be moderated by modifying the formula by which the guests receive their portions as the fourth root of their number - Nth guest receives N^0.25 percents of the remaining cake and not N% as suggested by the duke.
If we follow this recipe then the distribution of portions is as shown: 
This method gives a reasonable distribution of cake portions and meets the duke's wishes of serving larger portions to his important guests.  The portion sizes vary from 0.263% for guest 100 to a maximum of 1.576% for guest 13.  The cake remaining is 8.1% after all guests have been served.  

Wednesday, 21 February 2018

Undefined/Indeterminate Mathematical Operations Involving Zero and Infinity lead to fallacies and paradoxes

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I regularly receive emails and messages that claim to prove statements like 2 is equal to 1, 0 divided by 0 is 1 etc.  Such proofs almost always contain an indeterminate mathematical step (an operation where the the result is ambiguous) leading to paradoxical/fallacious conclusions.  
Another common error is to treat infinity as a number.  While infinity is a useful concept for indicating a limiting situation of increasingly larger numbers, it must not be treated as a number for mathematical operations.

Mathematics deals with numbers; each number has a well defined value or magnitude.  Manipulation of numbers is vital for our society to function efficiently - such manipulations follow well defined rules of addition and subtraction.  Results of such operations are unique and should have no ambiguity.  If there is ambiguity - the result is indeterminate, and for this reason they are unacceptable.
First let me explain why infinity () is not a number:
a.  If we think of infinity as a number that is larger than all finite numbers, then one can always think of a real number that is larger than that.
b.  Arithmetical operations do not apply to infinity - ability to add, subtract etc. is essential to the concept of a number.  For example;  if we write
          ∞ + 1 = ∞ + 2 =  
then it implies that 1 = 2 = 0, and also that infinity is a number that is larger than itself.
Similarly, if we write   1/∞ = 0 and also  2/∞ = 0 then it seems that 1 = 2 which is absurd.
c.  Greater-than, less-than, equal-to relations do not apply in the same way to infinity as they do to finite numbers.
d.  Infinity is a useful concept to indicate the limits to which the value of an expression approaches - for example, if x decreases from positive values towards zero then the value of 1/x increases, reaching the expression 1/0 at x = 0.  While 1/0 in indeterminate, the limiting value of 1/x as x gets ever close to zero is exactly definable.
                limx0(1/x= +∞     ≥ 0

The trend is shown in the slide

The above equation simply suggests that the limit, when x approaches zero, tends to infinity (an extremely large number) - it does not say that the value ever reaches infinity, rather that 1/x is increasing towards an extremely large positive value.
If x were changing towards zero from negative values then the limit will be written as 
                  limx0(1/x= -∞     ≤ 0

and the equation simply says that as x approaches zero, 1/x tends towards an extremely large negative value.  

Since, we can not treat infinity as a number, any mathematical operations involving infinities must be treated as 'not allowed'.  There might be special situations where one could consider infinity as a number and do maths with it, but we have to be very careful and watch out for paradoxical situations arising.  Hilbert's Infinite Hotel Paradox, Thomson's Lamp Paradox, 1 = 0.9999... are some well known examples.
Expressions (not a complete list) like 0 x ∞, ∞ + ∞, ∞ - ∞, p^∞, ∞^0 are indeterminate.

Infinite Series: Summing infinite series present some interesting situations.  If an infinite series is convergent then there is no problem, the nth term when n is very very large is going to be infinitesimally small and does not affect the sum in a material way. But what is the value of S for the infinite series:

S = 1 - 1 + 1 - 1 + 1 - 1 ...

We can organize the series in three different ways

S = (1 - 1) + (1 - 1) + (1 - 1) ...     = 0 + 0 + 0  ... = 0

S = 1 - (1 - 1) - (1 - 1) - (1 - 1) ...   = 1 - 0 - 0 - 0  ... = 1

S = 1  - S   or  2S = 1   which gives S = 1/2

Even though the first method has an infinite number of terms, in the second and third methods, we have one extra term. They are different series.

1 = 0.999...  :  This is my favourite fallacy.  Consider that 

                   x = 0.999999...            (eq.1)

three dots represent recurring nines to any large number (normally we say to infinity).  Multiply eqn. 1 by 10 on both sides

                 10 x = 9.999999...  =  9 + 0.999999...  =  9 + x          (eq.2)

From eq. 2;       10 x - x = 9 x = 9   or   x = 1

Therefore              1 = 0.999999

The problem with this type of proof is that the number of recurring nines in eq.1 is one more than in eq.2.  In eq.2, '0.999999...' is a different number from that in eq.1 and that creates the fallacious result.

Division by zero:   In mathematics, division is opposite to multiplication.  If a divided by b is equal to c, then c multiplied by b must be equal to a.  This rule does not work when we divide by zero.  
For example, let p = q/0; but p x 0 = 0 for all values of p (I shall deal with  the case of 0/0 later). There is no number p that, when multiplied by zero gives any other number except zero, therefore, it is fallacious to say that q/0 = p.  Dividing by gives a very large number in the limiting case when x0 but the limit when x = 0 is undefined.
Another way of looking at 'division by zero' is to consider division as a subtraction process - 24 divided by 6 is a subtraction process of taking away 6 sequentially until nothing remains.  The steps are;
24 - 6 = 18
18 - 6 = 12
12 - 6 =  6
 6 - 6  =  0
Four steps - 24 divided by 6 is 4.
When we divide 24 by zero, the steps will be as follows:
24 - 0 = 24
24 - 0 = 24 ...  for ever.  
The normal rules of division do not work when we divide by zero.
Zero divided by zero:  From our discussion above, 0/0 is not defined.  
We can look at 0/0 as follows:

zero divided by any number is zero - so 0/0 must be 0.
Any number divided by itself is equal to one - so 0/0 must be 1.
One can not have ambiguity in mathematical manipulations and the only conclusion we can draw is that zero divided by zero is undefined/indeterminate.

Zero multiplied by infinity:  If we start with the argument that any number, however small,  multiplied by ∞ gives infinity, that is,

               a x ∞ = ∞    then for a = 0, we obtain  0 x ∞ = 

However,  if limx0(1/x= ∞  then  limx0(x/x=   limx0(1) = 1

                                                   limx0(x^2/x) =  limx0(x) = 0
                                                                  limx0(x/x^2limx0(1/x) = 

Essentially, 0 x ∞ has no meaning in terms of mathematical operations.

Zero raised to the power zero (0^0): 

 What is the value of 0^0 ?  We know that any number raised to the power 0 is equal to one.  
Also we can multiply zero any number of times,  but we always get zero:

                    x^0 = 1  and  0^x = 0

these are valid mathematical operations.  However, in the limit, when x goes to 0, both expressions reduce to zero to the power zero - the first one is equal to 1 while the second one is equal to 0.
This is inconsistent with being an unambiguous result and for that reason unacceptable.  Zero to the power zero is indeterminate.

The Limit Paradox:  This is a paradox, I like very much.  Consider the equilateral triangle ABC.  All three angles of the triangle are equal to 60 degrees and the sides are the same length:

                               AB = BC = AC = a

D, F and E are midpoints of sides AB, BC and CA respectively. Therefore, triangle ADE and EFC will also be equilateral but sides of length a/2.

Now,                AB + BC = 2a = 2 x AC
also             AD +DE +EF + FC = 2 x AE + 2 x EC = 2 x AC  = 2a

We can continue to half the sides, and as shown above, the sum of the sloping sides will be equal to 2 times the base AC.

If we continue the process an infinite number of times then the sloping sides and the base coincide but according to our analysis the sum of all the sloping sides is twice the length of the base. This is a paradoxical result.

Again, the resolution is found in our concept of infinity.  The sloping sides are that way as long they are not horizontal - the height of the triangle is not zero.  then the angle of the tiny equilateral triangles formed remains at 60 degrees.  It collapses to zero as the sloping side coincides the horizontal base and in this limiting case - we do not have equilateral triangles any more - it is a different situation entirely. 

Final Word:  This publication was meant to discuss some indeterminate mathematical operations in a language accessible to non-specialists.  I have done away with formal statements as much as possible (I have not even used words like sets, axioms etc.) and for that reason, this blog piece may not be appreciated by the purist - but this is community education site.  
The main conclusions are: (a) Be very very careful when handling infinities -they are not numbers in the usual sense of the word; (b) While zero could be called a number, its position at the junction of positive and negative number lines makes it quite tricky to handle - again be very careful when doing mathematical operations with a zero.

Hope you enjoyed the excursion into mathematical paradoxes - let me know at

Sunday, 18 February 2018

Physics of Humidity, Relative Humidity, Health Implications of Low or High Humidity

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For the last two months, I have struggled to explain to my family the amount of water vapour present in the house.  We would have liked to kill dust mites in the house and for this, the level of relative humidity (RH) should be maintained at less than 45%.  But what does relative humidity of less than 45% mean? Searching many online resources, one could eventually find that it refers to a room temperature of 20°C.  And that is where the subject becomes confusing for the general public - for the same amount of water vapour in the air, RH is different at different temperatures.
In this blog, I shall go over the physics of humidity and discuss the importance of maintaining RH in the correct range (45 - 55%).  The physics is thankfully quite straightforward.  
The main constituents of air in a house are nitrogen, oxygen and water vapour.  The pressure of air is measured in millimeters of mercury or torr and for our purpose we can take it be fixed at 760 torr. Most of the 760 torr is due to the amount of nitrogen and oxygen present in the air.  At normal room temperatures, water vapour contributes only a small amount - called the partial pressure due to water vapour.  If we boil a kettle full of water then more water vapour will enter the surroundings; partial pressure of water and hence the humidity will increase. 
However, at any given temperature the air can only hold so much water vapour - it gets saturated - and any extra input of water (say boiling off from a kettle) will simply precipitate out and condense on the coldest surface in the room.  
So - at any temperature, it is only possible to have a maximum value of the vapour pressure of water - maximum humidity  - or as we say - the relative humidity is 100% at that temperature.  
The slide shows how the saturated vapour pressure of water (to create 100% RH) changes with room temperature:
It might be useful to list some saturated water vapour pressures (WVP) for 100% RH for temperatures (T) of interest to us:

              T (°C)      WVP (Torr)

          0              4.6
                4              6.1
                8              8.1
               12             10.5
               16             13.6
               20             17.5
               24             22.4
               32             35.9
               36             44.6

Notice that these values are much smaller than 760 torr.  At normal room temperatures the maximum amount of water that air can hold is not that much.

At T = 20°C , WVP is 17.5 torr - this is the maximum amount of water that the air at 20°C can hold and relative humidity (RH) is 100%.  
If the WVP is reduced to half this value, then the amount of water in the air is also halved and we say that RH is 50% at 20°C.

Consider the situation:  Outside temperature is 8°C and the air is quite damp - WVP or RH outside is 100% or 8.1 torr (see table).  If we open the windows wide and let the air in the room be replaced by the colder outside air, then RH in the room (assuming it stays at 20°C) will be reduced to 8.1/17.5 = 46.3%.  
If the room temperature has dropped to 16°C then RH = 8.1/13.6 or 59.6%.
The value of RH depends on both the amount of water in the air and the temperature.

Mass of water in the room:   Consider the contents of air at 760 torr.  The WVP is 17.5 torr -  remaining 760 - 17.5 = 742.5 torr is due to other constituents of air (mostly nitrogen and oxygen).
There is a law in physics that states that the weight of 22.4 litres of a gas is equal to its gram molecular weight at 760 torr and 0°C.  We shall use this law at other temperatures too - it is accurate enough for our purpose.  
The gram molecular weight of water is 18 gram while for air it is approximately 29 gram.
Therefore the weight of water vapour in 22.4 litres of air is 18*17.5/760 = 0.414 gram
Weight of 22.4 litres of air is = 29*742.5/760 = 28.3 gram
Amount of water = 0.414/28.3*1000 = 14.6 gram/kg for saturated vapour pressure. This is about 15 cc of water in 22.4 litres of air (1 litre = 1000 cc)  
For RH of 45%; Amount of water is 6.6 g/kg.  A level lower than 6.6 g/kg is the recommended level of water content at 20°C to kill dust mites.

We can now extend our table of WVP as follows:

   T     WVP    Water content in gram per kg of air
 (°C)    (torr)        Relative Humidity (RH)
                       100%   60%   45%    30%       

    0     4.6        3.9        2.3     1.7       1.2      
   4      6.1        5.1     3.1     2.3       1.5    
   8      8.1        6.8     4.1     3.1       2.1        
  12   10.5        8.8     5.3      4.0       2.7        
  16   13.6       11.4    6.8      5.1       3.4  
  20   17.5       14.6    8.8      6.6       4.4         
  24   22.4       18.8   11.3     8.5       5.7        
  32   35.9       30.0   18.0    13.5      9.0  
  36   44.6       37.3   22.4    16.8     11.2          

I have highlighted in red the 'recommended' value of RH at different temperatures. 
Water content of 6.6 g/kg is easy to achieve for room temperatures of 20°C or less.  At higher room temperatures, common in the summer and in tropical climates, RH will have to be less than 30% to kill dust mites.  Interestingly, water content levels of 6 to 8 g/kg are recommended for healthy living.  These levels are also suitable for dust mites, many fungii and molds to reproduce and grow. 

To complete our discussion, let us calculate the amount of water in a normal size room - say 5m x 4m x 2.5m  or 50 m^3.  
The density of air is 1.225 kg/m^3 
The mass of air in the room is 50 x 1.225 = 61.25 kg
Mass of water at 20°C and 45% RH = 6.6 gram/kg x 61.25 kg = 404 gram
An average size of room contains, at 20°C,  about 400 gram or 0.4 litres of water at RH of 45% or about 8 cc water per m^3.  This is not a lot of water.  We can increase RH in a room by boiling some water in a pan or electric kettle. As a rough guide, 0.1 litre water, when boiled off, will increase humidity by 10% in a 50 m^3 space at 20°C.

How is humidity measured:  It is common to use a digital hygrometer which displays the temperature and relative humidity of the surrounding air.  A hygrometer calculates the humidity by measuring the capacitance or resistance of the element.
A capacitor has two metal plates with air in between them,  It is used to store electric charge - its capacity to store charge is affected by the amount of water vapour between the metal plates. Measurement of the capacity provides an accurate value of the humidity.
A resistive sensor is generally a piece of ceramic that is exposed to surrounding air.  The humidity of the air in the ceramic resistor affects its resistance and hence the current flowing in it when connected to a battery.  

Adverse effects of too high or too low humidity: 

Low Humidity:  In winter months, it is quite likely that the water content in your house will be less than 7 g/kg; with good ventillation, it could quite easily drop to 4 or 5 g/kg.  Remember, it is the water content that is meaningful, relative humidity numbers depend on room temperature and are less useful.

The low humidity air can lead to dry skin, itchy/dry eyes, irritated sinuses and throat.  A hygrometer is the best way of monitoring humidity in the house, but tell-tale sign of houseplants drying out, wallpaper peeling at the edges or static electricity point to low humidity conditions. 
Exposure to low humidity can dry out and inflame the mucous membrane lining of the respiratory tract increasing risk of infections like cold and flu. In low humidity environment some viruses may be able to survive longer, further increasing the risk of infection.

High HUMIDITY interferes with THE BODY'S Cooling Mechanism:  Human body works best when the core temperature is 37°C.  When outside temperatures approach 37°C, the body’s thermal regulation system attempts to cool it by transporting heat from the core organs by increasing blood circulation to the skin and sweating. Sweating, one of the main cooling mechanisms of the body, works by evaporating water that is excreted through the skin. This is where humidity becomes important. The concentration of water in the air (humidity) determines the rate at which water can evaporate from the skin. When the humidity in the air is high, it is not able to absorb the extra moisture  from the sweat. The result is that sweating, instead of giving any relief, makes us feel hot and sticky. High humidity makes us feel hotter, more uncomfortable and unable to lose heat our core temperature actually begins to rise.  The  body compensates by working harder to cool us down. The loss of water, salt and chemicals can lead to dehydration and chemical imbalances within the body leading to heat exhaustion.  
The heat index chart tells us quantitatively that in high humidity conditions the body feels hotter than the actual ambient temperature. For example, for an ambient temperature of 104°F (40°C) and a relative humidity of 40%, the water content in air is about 20 g/kg and it will feel like 119°F (48°C). But if the relative humidity increases to 55% (water content = 27 g/kg) the temperature will feel like 137°F (58°C)! 

At high water content (greater than 7 g/kg) fungus, molds and dust mites also survive and become a problem.  As I had mentioned earlier, for high ambient temperatures, it is not possible to reduce water in the air to less than 7 g/kg and one might need to use dehumidifiers. Maintaining a dehumidifier in a clean condition is another issue ...    

Monday, 12 February 2018

Letter Frequency in Spellings of Words and Numbers in the English Language

Index of Blogs

The subject of this blog is completely different - I think it is insane.  
But I had to write it down as it appears so fascinatingly interesting, albeit useless.

If you look at the website (, you can find the order in which letters of the alphabet occur in words of the English language.  They occur in the following order - highest frequency first:

e t a o i n s r h l d c u m f p g w y b v k x j q z
The first 12 letters are found in 80% of the words. 
Actual values are (notice slight discrepancy after letter m)

The story begins with my granddaughter writing to me to say that the spellings of numbers from zero to ninety-nine do not contain the first four letters of the alphabet, namely a, b, c and d.  I was surprised to see the letter a in the list as it is the third most frequent letter used in English language, and to be missing in the spellings of the first thousand numbers (it first appears in a thousand) would be curious.
I then got down to prepare a list of letters missing in number spellings. What use is it? - I have no idea but I think it is insanely interesting.
(The notation used here is: 10^n is 1 followed by n zeros; 10^2 is 100; 10^6 is 1,000,000 or 1 million; and so on)
       Letter                   First Appearance 

          a                 10^3  or 1,000  Thousand
          b                 10^9       or       Billion
          c                 10^27     or       Octillion
          d                 10^2       or       Hundred 
          j                 does not occur in any spellings
          k                does not occur in any spellings
          m                10^6       or       Million
          p                 10^24     or       Septillion
          q                 10^15     or       Quadrillion

I might have missed something and got one or more errors in the list - please let me know.

I could start looking at negative powers of 10 but I think that is taking things a bit too far.

I hope you enjoyed reading through the blog - slightly different from the usual serious stuff; this is what you get when you start talking to your grandchildren.