A Couple of Challenging Geometry puzzles - Created by Martin Gardner - Recreational Mathmatics

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These Puzzles are mathematically challenging

I was inspired to look into Martin Gardner's (1914 - 2010) work by an article in the recent Physics World issue.  Gardner ran a column in Scientific American for 25 years and published books on recreational mathematics.  Some of his puzzles can take a life of their own and result in a few sleepless nights.

I have chosen two examples from the famous collection of his puzzles based on geometry.  I use these puzzles to show how trigonometry and geometry are equally efficient in solving problems.  Personally, I find trigonometry easier to use - it provides ready-made relations between the sides and angles of a triangle which might be more difficult to visualize in a geometrical context - particularly if you are using hand-drawn diagrams that may not be  constructed accurately.

The first problem that I have chosen is best illustrated by the following diagram (click on the figure to see a full page image)

I shall give the geometry based solution but let us first try the easier trigonometric solution of the above problem:

Geometrical Solution: If we look at the angles at the corner marked T in the diagram:

angle WTR = angle TRS = angle C
angle WTP = angle TPS = angle A

and angle WTR = angle WTP + angle PTR
or          C = A + angle PTR 

Essentially, to solve the puzzle, we need to prove that angle PTR = angle TQS 

One way to do this is to complete a right angled triangle that will include angle PTR.  This may be done by drawing the perpendicular PV from point P on an extended line TR - this gives us the following figure:

Line PV has been drawn parallel to the diagonal YR of the middle square to ensure that angle PVT is 90 degrees.  In the figure above, we can clearly see that right angled triangles QST and TVP are similar as all their sides are in the same ratio (1 :√2), viz. 

QS  =  2,       ST  =  1     and   TQ  =  √5

TV  = 2√2,    VP = √2     and   PT = √10

This means that corresponding internal angles are equal.

 Therefore, angle facing side ST (angle B)  is equal to angle facing side PV  or  

         angle PTV  (or PTR)  =  angle B

Thus we can be sure that angle PTR = angle B, and 

                        angle C = angle A + angle B

Notice, we have used the properties of similar triangles to solve the above puzzle.  This method is used frequently in solving geometry problems and sometimes it can provide a quicker and more concise solution. To demonstrate this point,  I discuss another geometrical puzzle in the following:  

PUZZLE No. 2:

There are three different ways that one can solve this problem.  
Geometry, Trigonometry and Pythagoras Theorem. 
We shall try all three methods:

Solution using Geometry:  Here we have three similar right-angled triangles - their internal angles are the same.  It is straightforward to show that

Triangle ABC:  angle BAC = A, angle BCA = C and angle ABC = 90
Triangle BDA:  angle BAD = A, angle ABD = C and angle ABC = 90
Triangle BDC:  angle CBD = A, angle BCD = C and angle BDC = 90

In similar triangles, the ratios of corresponding sides are equal. This implies that

For angles 90 degrees and A ___  AC:BC = BC:DC = AB:BD
   or   25:a = a:16 = c:H
   or     a^2 = 25 x 16 = 400   or   a = 20 cm

For angles 90 degrees and C ___  AC:AB = BC:BD = AB:AD
  or   25:c = a:H = c:9 
  or     c^2 = 25 x 9 = 225    or   c = 15 cm

We also have a:H = c:9   or  20 x 9 = 15 x H   or  H = 12 cm

Therefore   H = 12 cm,  a = 20 cm and c = 15 cm

Solution using Trigonometry: 

In right angled triangle ABD,   tan A = BD/AD = H/9
In right angled triangle CDB,   tan C = BD/DC = H/16

But angle C = 90 - angle A;   therefore  tan C = tan (90-A) = 1/tan A

This gives us ---   tan A = H/9 and 1/tan A = H/16
On multiplying the two equations:  1 = H^2/(9x16)

            or   H^2 = 144   or H = 12 cm

Solution using Pythagorus Theorem:  

In the three triangles:

     a^2 + c^2 = 25^2     or   a^2 + c^2 = 625    eq.1 
     H^2 + 16^2 = a^2    or   a^2 - H^2 = 256     eq.2  
     H^2 +  9^2  = c^2    or   c^2 - H^2 =  81      eq. 3

Subtract the sum of equations 2 and 3 from equation 1 to obtain

2 x H^2 = 625 - (256 + 81) = 625 - 337 = 288

or H^2 = 144   or   H = 12 cm

Final Word:  Geometrical puzzles are really addictive.  I remember during my school days, my father had asked my younger brother and I to see if we can divide an angle in three equal parts just using a compass and ruler.  Of course, this is not possible but we spent weeks of our play time trying to work it out and used up a large amount of paper.  But it was lots of fun and we learnt many useful tricks and theorems in mathematics during this time.  It was one of the most productive period in term of return from our unsuccessful attempts. Struggle makes you wiser and more resourceful. 

Keep puzzling...

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.

                      168 x 169/2 - (2P + 1) = 14001
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 on a page, the number on its front page face is 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 ektalks@yahoo.co.uk or by leaving your comment on the blog site.    

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 N 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 99 to a maximum of 1.576% for guest 13.  The cake remaining is 8.1% after all guests have been served.  

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.
                limx→0(1/x) = +∞     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 
                  limx→0(1/x) = -∞     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 x gives a very large number in the limiting case when x→0 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 limx→0(1/x) = ∞  then  limx→0(x/x) =   limx→0(1) = 1

                                                   limx→0(x^2/x) =  limx→0(x) = 0
                    
                                                                  limx→0(x/x^2) = limx→0(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 ektalks@yahoo.co.uk

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

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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 (http://letterfrequency.org/), 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:

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.