Some Fun With Algebra How good is your counting?

[Omkar™ 0]

A matter of $1...

 

Once A, B & C went to a restaurant. After having tea and light refreshment they asked for bill, the amount of bill was $30.

They decided to pay the bill amount on equal share.

Everybody took out $10 from their pocket and paid the bill.

They were yet sitting and gossiping in restaurant,

Suddenly the waiter realized that the amount of their bill was $25 and not $30.

He took $5 from counter and decided to return it to them.

Then he thought they were 3 how will they share $5 ?

So he decided to keep $2 to himself and return only $3 to them.

 

He came to their table and returned them $3 with apology.

Everybody took $1 and put in their pockets.

 

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Now my question is first time everyone paid $10

Later they get $1 refunded.

So everybody paid $9.

 

$ 9 * 3 = $ 27 ..... 1

 

The waiter put $2 in his pocket.

 

$ 27 + 2 = $ 29 ..... 2

 

So...... Where is the remaining $1 ????

 

Start thinking, and soon you'll return to the age when you were taught how to add using your

 

fingers!

 

-Omkar Ekbote

[yeh 0]

Hehe.. nice! :DThe part 27+2=29 does the trick of misleading people into the wrong thinking path. The waitress should have only 2 becoz 27-25=2. What the waitress get is the difference between what A, B, C paid and the actual price. After all, she pockected some money from them. :PIf somehow the bill now becomes 30, which is the original price, A, B, C would then take out 1 each and the waitress would take out 2. 1+1+1+2=5 and 25+5=30. Hurrah!

[cyborgxxi 0]

Very Nice. I love a good puzzle. Anyway, here's another puzzle for you braniacs out there. Answer this one.Find the next three terms in this sequence: 3, 3, 6, 8, 15, 24.....

[Logan Deathbringer 0]

24?

[apurva 0]

its 39.. its a fibonacii series..

[miCRoSCoPiC^eaRthLinG 0]

its 39.. its a fibonacii series..

 

You're quite wrong there Unitechy.. While this ressembles the Fibonacci Series, this doesn't progress like that beyond the first 3 numbers in sequence. An ideal Fibonacci Series would sequentially progress as the sum of the last two numbers. For example: 0, 1, 1, 2, 3, 5, 8.....

 

Here the series progresses a bit differently: 3, 3, 6, 8, 15, 24.....

 

Admittedly, judging by the first 3 in the set, it looks like Fibonacci.. but starting from 4th number it takes a different turn. You'd notice a that from the 3rd term, apart from the sum of last 2 numbers, the term gets padded with a sequence that progresses like 0, -1, +1, +1 ...

Let me clarify this a bit further... (if we start by considering the 3rd term)..

3 + 3 + 0 =6

6 + 3 - 1 = 8

8 + 6 + 1 = 15

15 + 8 + 1 = 24

 

This is where it gets me baffled... and I can't crack the series being padded to the original sequence.. Maybe an example of the 7th number would help me get at it better... though I have a feeling this series would alternate between 0, -1, +1, +1, -1, 0 ... like that... if that's the case, the 7th term would be..

24 + 15 - 1 = 38

[pyost 0]

Well, m^e, I've had the same thing as you in mind. However, that way it is quite hard to predict what will go next.

 

We have:

+0

-1

+1

+1

 

There are numerous possibilities and we can't find what is being repeated. But here's a guess: +1, +0, -1, +1. It is quite stupid, but that way we would get this sequence:

 

3 3 6 8 15 24 39 62 102

 

I might have made a mistake when calculating, but this is one of the possibilities.

[apurva 0]

hmmm.. might be true.. but its quiet confusing..

at one tme -1 then +1

 

 

You're quite wrong there Unitechy.. While this ressembles the Fibonacci Series, this doesn't progress like that beyond the first 3 numbers in sequence. An ideal Fibonacci Series would sequentially progress as the sum of the last two numbers. For example: 0, 1, 1, 2, 3, 5, 8.....

 

Here the series progresses a bit differently: 3, 3, 6, 8, 15, 24.....

 

Admittedly, judging by the first 3 in the set, it looks like Fibonacci.. but starting from 4th number it takes a different turn. You'd notice a that from the 3rd term, apart from the sum of last 2 numbers, the term gets padded with a sequence that progresses like 0, -1, +1, +1 ...

Let me clarify this a bit further... (if we start by considering the 3rd term)..

3 + 3 + 0 =6

6 + 3 - 1 = 8

8 + 6 + 1 = 15

15 + 8 + 1 = 24

 

This is where it gets me baffled... and I can't crack the series being padded to the original sequence.. Maybe an example of the 7th number would help me get at it better... though I have a feeling this series would alternate between 0, -1, +1, +1, -1, 0 ... like that... if that's the case, the 7th term would be..

24 + 15 - 1 = 38

 

[mediafork 0]

Heh.. Nice problem there... I think I've heard it before but it's always cool to hear these brain teasers again every now and then..Very mind boggling!Got anymore cool brain teasers for us?

[vizskywalker 0]

Well, except for the 8, it is each number TN where T stands for Term and N is the term number is 3 * FN where F is the Fibonacci term and N is still the same term number. Could it be a typo?~Viz

[Logan Deathbringer 0]

I'd like to see the answer...I still stand by my original post though. I hope he posts the answer. lol

[iGuest 3]

I've got it!

In case you don't want me to reveal the answer, made it so you need to click the link below to show the answer.

Unitechy was almost correct by suggesting that it may be a Fibonacci sequence. For those of you who don't know what a Fibonacci sequence is I have included some info about it below, as well as some info about Fibonacci that I hope you will find interesting.

The 8 is a typo; it should be a 9. The sequence should read:
3, 3, 6, 9, 15, 24
[spoil]If you divide each term in the sequence by 3 you get
1, 1, 2, 3, 5, 8
A Fibonacci sequence!

The next term in the Fibonacci sequence is 13, so the answer to the puzzle is 13 X 3 = 69.

I notice that vizskywalker also got it right, although the explanation they gave wasn't very clear.[/spoil]

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Fibonacci and the Fibonacci sequence

In this section of this post I will be writing about Fibonacci, the Fibonacci sequence, the decimal system and the golden section.

Let's start with the Fibonacci sequence.
The Fibonacci sequence is a sequence in which any given term (except for the first two terms) is equal to the sum of the two previous terms. For example:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55...
As you can see:
1 + 1 = 2
1 + 2 = 3
2 + 3 = 5
3 + 5 = 8
5 + 8 = 13
8 + 13 = 21
13 + 21 = 34
21 + 34 = 55
Fibonacci sequences can begin with any two numbers, but usually start with 1, 1.

We'll come back to Fibonacci sequences later when we are looking at the golden section, but for now lets have a brief look at Fibonacci and his achievements.

Fibonacci was born in Pisa in c1175, and never actually called himself "Fibonacci". He was born "Leonardo Pisano" (Italian for "Leonardo of Pisa"), but reffered to himself as "filius Bonacci" (meaning "son of the Bonacci family") and "Leonardo Bigollo" ("Leonardo the traveler"). "Fibonacci" is a shortening of "filius Bonacci" and is how Leonardo Pisano is best known in modern times. What I believe to be Fibonacci biggest achievement is the introduction of the decimal number system to Europe. This system used the Arabic symbols 1, 2, 3, 4, 5, 6, 7, 8, 9 and 0. The decimal system is a base-10 system. The reason for using a base-10 system is that when the decimal system was introduced counting using the fingers was common practice (ten fingers, base-10). The word "digit" comes from latin for "finger".

Although the Fibonacci sequence bear his name, it is thought that he probably did not invent it.

And now, finally, we will take a very brief look at the golden section. The golden section is also known as the golden ratio, the golden mean, the divine proportion and Phi (Φ). The golden section is an irrational number (a number with an infinite number of decimal places which don't repeat) and ≈ 1.618. The golden section frequently occurs in nature, geometry and trigonometry.

And that concludes this brief look at Fibonacci. If you would like to learn more, I have included a few links below:

Multi-award winning Fibonacci page on the Surrey University site:
http://forums.xisto.com/no_longer_exists/

Biography of Fibonacci on St. Andrews University site:
http://forums.xisto.com/no_longer_exists/

[iGuest 3]

NOTE: While we are on the subject of Fibonacci, I thought I might just make this post. I realise that Fibonacci is slightly off topic (as this thread is for puzzles), so I have included a puzzle at the end of this post in order to bring it back on topic. The reason I have posted this as a separate message from my previous one rather than adding it on to the end of it is that I feel what is discussed in this post is slightly different from what was discussed in my last post, and if I simply appended the information contained within this post onto my previous post then it could become quite confusing. I am thinking of this post as an article, one which I may even reproduce on a hosted webpage. Now, if you are wondering why I am posting an article about Fibonacci in a thread about algebraic and mathematical puzzles, there are two reasons; the first is that I did not know where else to post this article, and the second is that this thread was already on the subject of Fibonacci. I will now proceed with the article.

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After reading Dan Brown's 'The Da Vinci Code', I was shocked to find him claiming that the Fibonacci sequence, and the divine proportion (Phi) were evidence of God's creation. I am an atheist, and would like to disprove those claims and argue that "Phi's ubiguity in nature" does not proof that God exists. I will look at some of the examples of Phi appearing in nature given in 'The Da Vinci Code' and try to find rational explanations for them. I will point out where Brown got it wrong (as well as where he got it right) and will hopefully increase your understanding of Fibonacci.

 

First of all, let's look at one of the first statements Brown makes about the Fibonacci sequence:

"Mathematician Leonardo Fibonacci created this succession of numbers in the thirteenth century."

This one sentence is full of incorrect information:

Fibonacci was not called "Leonardo Fibonacci"; his real name was "Leonardo Pisano" (see previous post for more info).

Fibonacci did not create the Fibonacci sequence. Fibonacci wrote about this sequence in his book 'Liber Abaci', but probably heard about this sequence from his contacts in India. The Fibonacci sequence was first discussed by Indian scholars.

The Fibonacci series was not created in the 13th century; Fibonacci published 'Liber Abaci' in 1202, but Indian scholars had discussed it much earlier; Pingala in the 5th century BC, Virahanka in the 8th century AD, Gopala in c.1135 AD and Hemachandra in c.1150 AD. The Fibonacci sequence was given the name "Fibonacci sequence" by Edouard Lucas (the mathematician who invented the Lucas Series) in the 19th century.

For more information about Fibonacci, check out these links:

Ron Knott's Fibonacci biography

St. Andrew's University Fibonacci biography

 

 

The number Φ (Phi) also features in 'The Da Vinci Code'. Brown says that Φ = 1.618, but this is not exactly true. Φ ≈ 1.618 (≈ means almost equal to). The mathematical definition of Phi is a number equal to 1 plus the square root of five, divided by two. If you take any two consecutive numbers in the Fibonacci sequence or the Lucas series (a series of numbers closely related to the Fibonacci sequence) and divide the greater value by the lesser value then you get a number approaching Φ (if you divide the lesser value by the greater value you get the number φ, which is equal to Φ-1). In the book, Brown states that:

"Phi's ubiguity in nature clearly exceeds coincidence, and so the ancients assumed the number Phi must have been preordained by the Creator of the universe."

 

He also states that:

"Plants, animals and even human beings all possess dimensional properties that adhere with eerie exactitude to the ratio of Phi to 1."

before going on to give a list of examples. I will now try to give reasons for some of these.

Brown claims that if you divide the number of female bees by the number of male bees in any beehive in the world, you always get the number Φ. This is true, and it is not simply coincidence, but neither is it proof that God created Earth. There is a logical, and interesting, explanation for this; and it is all to do with the way in which honeybees reproduce. Honeybees are parthogenic. The queen mates only once in her lifetime, retaining the sperm within her. She can use this sperm anytime she wants to lay fertilised eggs, but can also choose to lay unfertilised eggs. Fertilised eggs will hatch into females, unfertilised eggs will hatch into males. This means that a female bee has 2 parents (1 male and 1 female), and a male bee has 1 parent (1 female). If we look at the grandparentage of a bee, we can see that a female would have 3 grandparents (1 male parent means 1 female grandparent, and 1 female parent means 1 male grandparent and 1 female grandparent), and a male honeybee would have 2 grandparents (1 female parent means 1 female grandparent and 1 male grandparent). This information might not sound very startling at first, but look what happens when put it in a table:[tabl]

Gender of

Honeybee

????[/tab]

Number of

Parents

????

Number of

Grandparents

????

Number of

Great-grandparents

????Number of

Great-great-

grandparents

????Number of

Great-great-

great-grandparents

[tab]Number of

Great-great-great-

great-grandparents

Male[/tcol]1235813

Female23581321[/tabl]

As you can see, both of these (1,2,3,5,8,13 and 2,3,5,8,13,21) are examples of the Fibonacci sequence. Notice also that for any given generation: the number of ancestors of that generation a male bee has is a Fibonacci number, and the number of ancestors of that generation a female bee has is the next consecutive Fibonacci number. Phi is equal to a Fibonacci number divided by the previous Fibonacci number. The number of males and females in any given generation are consecutive Fibonacci numbers, thus the number of females divided by the number of males is equal to Phi.

Brown claims that on a Nautilus shell the ratio of each spiral's diameter to the next is Φ. Spirals in which this occur are called golden spirals. The pattern on the shell of a Nautilus is a logarithmic spiral (a spiral in which the distance between spirals increases with every rotation, but the ratio of each spirals diameter to the next remains the same) but not, as Dan Brown would have you believe, a golden spiral.

Brown claims that sunflowers seeds grow in opposing spirals, with the ratio between the the diameters of the larger rotation and the smaller rotation being equal to Phi. This is true, but there is a very simple reason for this. Here is a quote from the Golden Ratio site:

"The reason this spiral is present seems to be that it forms an optimal packing of the seeds so that, no matter how large the seedhead, they are uniformly packed, all the seeds being the same size, no crowding in the centre and not too sparse at the edges." In fact, a lot of things grow in golden and logarithmic spirals because it is the optimum way for them to do so (and not because God made them that way).

Brown claims that "the human body is literally made of building blocks whose proportional ratios always equal Phi." This claim is subjective and unsubstanciated. The human body contains so many joints and possible points of measurement that finding Phi is quite easy. If you are actively looking for Phi in objects then the chances are that you will find it eventually (someone even claimed to have found it in a door). This is also true of most ratios. Don't believe me? Try it out for yourself. Here are some examples:

 

[tabl]

Ratio of 1:2 - ratio of the number of people an individual is to the number of biological parents he/she has- ratio of how many sight organs a person has to how many tasting organs a person has- ratio of the area of a semi-circle to the area of a circle

Ratio of 3:1 - ratio of the number of molars in an adults mouth to the number of caninesratio of the circumference of a circle to its diameter as implied by 1 Kings 7:23

Ratio of 7:8 - ratio of the length of my thumb to the length of my outermost finger.[/tabl]

For more information on Phi in nature, check out these links:

Golden Ratio site

An article on mathforum.org

An article pointing out some of the mathematical errors Dan Brown made in 'The Da Vinci Code'

 

I hope I have managed to enlighten you a little, and convince you that Phi is not a sign from God. 'The Da Vinci Code' is a really great read, despite the factual inaccuracies, and my intention in writing this article is not to discourage you from reading the book; my intention is to encourage you to question everything. There have been many times throughout history when the very foundations of science have been shaken by controversial new theories which have been proven to be true. We are living in an age where it is becoming increasingly easier to access information, so there is no excuse for subscribing to ideas for which there is evidence to suggest are flawed.

 

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Depending on the popularity of this post, I might do some more "disproving 'The Da Vinci Code'" articles or some Fibonacci articles. Also, I plan on writing an article later in which I will go a long way towards disproving God, and will probably reference this article to show that there is no evidence of God in Nature.

 

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And now for a puzzle, to bring this thread back on-topic:

 

This is a puzzle by Henry E. Dudeney; it first appeared in his book '536 Puzzles & Curious Problems' in 1967. This puzzle was adapted from Fibonacci's 'rabbit problem'. Click the link below the puzzle to reveal the answer:

 

If a cow produces its first she-calf at age two years and after that produces another single she-calf every year, how many she-calves are there after 12 years, assuming none die?

NOTE: A "she-calf" is only considered a "calf" while they are younger than 1 year old.

[spoil]

We start off with one she-calf at age 0.

After one year, we have 1 cow who is one year old and 0 births.

After two years, we have 1 cow who is at least two years old and 1 birth.

After three years, we have 1 cow who is at least two years old, 1 cow who is one year old and 1 birth.

After four years, we have 2 cows who are at least two years old, 1 cow who is one year old and 2 births.

After five years, we have 3 cows who are at least two years old, 2 cows who are one year old and 3 births.

After six years, we have 5 cows who are at least two years old, 3 cows who are one year old and 5 births.

After seven years, we have 8 cows who are at least two years old, 5 cows who are one year old and 8 births.

After eight years, we have 13 cows who are at least two years old, 8 cows who are one year old and 13 births.

After nine years, we have 21 cows who are at least two years old, 13 cows who are one year old and 21 births.

After ten years, we have 34 cows who are at least two years old, 21 cows who are one year old and 34 births.

After eleven years, we have 55 cows who are at least two years old, 34 cows who are one year old and 55 births.

After twelve years, we have 89 cows who are at least two years old, 55 cows who are one year old and 89 births.

You should notice that the number of births in each year are part of the Fibonacci sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89

[/spoil]

 

Please post any other puzzles you have below.