How many squares are there on a chessboard?

Simple answer. 64 right? This is the thought problem I gave my year 9 class for their end of Friday problem solving half hour.

On the face of it 64 is the right answer. there are 64 squares on the board

But what about the 2x2 squares and the 3 x 3 squares.... and so on? Apart from one girl who finished the task in around five minutes who's explanation was to point to the board and say " Can't you see them?" ( I think we have a genius there!) the class attacked it with gusto. One or two piled in to systematically counting them all ( We'd just finished a lesson on listing combinations systematically then finding probabilities so I blame myself) 

A more interesting line of thought was looking for patterns on the board and linking them with square numbers. There were some excellent visual examples of this from the class and this is my less than stellar example

And this is the breakthough for solving the problem. There is a link between the square numbers and the number of squares on a chess board. There are

1, 8x8 square
4, 7x7 squares
9, 6x6 squares
16, 5x5 squares
25, 4x4 squares
36, 3x3 squares
49, 2x2 squares
64, 1x1 squares

Therefor there are 204 squares on a chessboard.  

There is a formula for the sum of squares of the integers 1^2 + 2^2 + 3^2 + ...  + n^2

                   n(n+1)(2n+1)
         Sum  = ------------
                         6

In our case, with n = 8, this formula gives 8 x 9 x 17/6 = 204. Following the pattern we could find how many squares are on any subdivided square. For example, a 9x9 square would produce 285 squares

The interesting thing is that all the answers from the pupils came with the same error in arithmetic(211) I'm still trying to work out why this is the case. Any thoughts on this point will be gratefully received.

Is a Toblerone a triangular prism? - Maybe

I started this blog in a flourish of enthusiasm when I started going into the classroom...Then reality and exhaustion kicked in and i've been looking for a way to restart this ever since. So, given I have a lesson observation tomorrow now seems like the perfect time!

So there I was reviewing the papers on Radio Humberside and, as is now becoming customary, the host Andy Comfort blind sides me with a maths related question.* What is the correct name for the shape of a Toblerone? Apparently there had been some debate as to whether it was a prism or triangular prism. Naturally, and without thinking, I said it was a triangular prism. Then on the way home from the studio I got thinking. This is not a prism. (credit)

A triangular prism is a prism with a continuous triangular cross section. On this definition, each individual piece is a triangular prism (if cut correctly), as is the box but clearly the whole bar is not.  The best I can come up with is each prism is connected by a truncated ovoid therefor, is a Toblerone a prism?

It depends what you are talking about but the bar itself is not. Its a series of triangular prisms connected by truncated ovoids.

Now, back to the lesson planning!

* So far i've managed bits of calculus 3D shapes, multiplications and various bits of numeracy.... One day I'll have a brain fart and embarrass myself for all time and lose all confidence of parents!

Perfect numbers

Last month I wrote about happy numbers and sexy numbers so I thought I’d complete a little hat trick and talk all about perfect numbers.

A number is said to be perfect if it equals the sum of its divisors. The second perfect number (28) is divisible by 1,2,4,7,14. 1+2+4+7+14=28

Perfect numbers are rare. At the last count only 48 have been discovered. It is not known whether there are an infinite number of perfect numbers or that if there are any odd ones (Every one found so far is even) and these remain open questions to be solved

Perfect numbers are also entwined with prime numbers as every even perfect number can be represented by the form 2^n − 1(2ⁿ − 1), where 2ⁿ − 1 is a prime number (otherwise known as a Mersenne prime). In order for 2ⁿ − 1 to be a prime number n must be a prime number. This is known as the Euclid–Euler theorem.

Just to get my retaliation in early this time there are no uses whatsoever for perfect numbers (It doesn’t mean one won’t be found, prime numbers were nothing more than a curiosity for centuries) but who says everything we do in life has to be utilitarian? Let’s just marvel at the beauty, interesting nature and perfectness of a small list of numbers.

Happy numbers

A couple of weeks ago I talked about sexy numbers but what makes a number happy? To find out if a number is a happy number you go through the following process.

Pick a number (Lets say 23) square each digit and add them together. so 4+9=13. Now repeat the process.

1+9=10
1+0=1

We've taken a number and through this procedure we have reduced it to 1 so it is now happy. If the number never reaches one then it is an unhappy number. Poor thing :-(

We can also have happy prime numbers such as 313, 331, 367, 379. these numbers are also interesting for another reason to this number geek. There are many happy primes before these numbers but in the Doctor Who episode 42 the numbers 313, 331, 367, 379 are used for unlocking a sealed door but no one can spot the pattern except The Doctor.....see, you never know when this trivia may come in useful!

The day politicians legislated break maths

Just about everyone who remembers their schooling can recall pi to a couple of places (3.14159......) In reality, though, as an irrational number it never ends and never settles into a repeating pattern. (if you have the time you can see the first million digits of pi here) I say in reality, but there was an attempt to define it and stop all the messiness with lots of decimal places and make a law to replace it with 3.2 - much tidier, but just the slight drawback of being mathematically incorrect!

In 1897 the general assembly of the state of Indiana was persuaded by a Doctor by the name of Edwin J Goodwin to consider his idea that Pi was not this messy irrational number but 3.2. As well as producing an incorrect calculation of pi, Mr Goodwin graciously said that schools in Indiana could use his idea free of charge and the state could even share the royalties when he sold his idea to to schools in other states. All the state had to do to share this windfall was make pi=3.2 the law of the state. After being bounced around various committees including the committee on swamp lands (the mind boggles) the bill was passed without any objections.

So, the bill went up to the state senate for ratification where after its first reading it was sent to the committee on temperance to study the detail and also gave it the thumbs up. (Politicians on a temperance committee believing dodgy maths, that's a first!) To the good fortune of the state's school children a professor of mathematics happened to be in the state capitol on other business and was shown the bill. The good professor took it upon himself to teach the state senators some elementary mathematics and the state senate decided to indefinitely postpone the bill. So,somewhere in some dusty filing cabinet there is still a bill marked pending just waiting for the next set of politicians who think they can legislate for mathematical truth!

The moral of the story? Well, lets leave that to Neil Degrasse Tyson

199th day of the year - A very interesting number

For those who follow me on other social media platforms you’ll have seen my musings on why 197 is an interesting number. Well, as it turns out 199 is also an interesting number.

199 is a twin prime (Twinned with the aforementioned 197) Twin primes are prime numbers that are separated from each other by two….Not as exciting as sexy primes but mathematically more challenging!

It is an 'invertible prime.’ If you turn the number upside down it reads 661 which is also a prime number.

It’s also a ‘circular prime.’ It stays a prime number no matter how you arrange its digits (919 and 991 are also prime numbers)

199 is the sum of 3 consecutive primes 61 + 67 + 71 (Incidentally 661 is also the sum of three prime numbers 211 + 223 + 227)….And also the sum of 5 consecutive primes 31 + 37 + 41 + 43 + 47

Aaaaand if you add its digits up the sum is 19 which is also a prime number.

Exciting eh! :-)

-------------------------------

Oh go on then since I didn’t print it here, this is what is interesting about 197

As well as being a prime number 197 is the sum of the first twelve prime numbers: 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37

197 is also the sum of all digits of all two-digit prime numbers: 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Aaaand....197 is the smallest prime number that is the sum of 7 consecutive primes: 17 + 19 + 23 + 29 + 31 + 37 + 41

Pythagoras the murderer!

Picture the scene. It's around 500 BC and it’s a lovely Summer day off the coast of Greece. Out on the water though there is a kerfuffle. A man is fighting for his life after getting thrown over the side of a boat and left to die. His name is Hippasus of Metapontum and he had revealed a basic piece of mathematical logic.

Let us dial back a bit and talk about Pythagoras. Most people remember Pythagoras from school and even if they can’t remember what his theorem is about most remember it’s something to do with triangles. (for a nice visual recap see here). Pythagoras was much more than a developer of mathematical theorems though, he was also the leader of a full on religious cult which believed in vegetarianism and that numbers were divine.

One of the central doctrines of the Pythagorean cult was that all numbers could be written as fractions. Hippasus was pretty certain he had seen numbers that didn’t fit this pattern and mentioned this to Pythagoras who, according to some legends, suggested a little ride in his boat and, well, you now know the rest.

So how can we can use (and maybe Hippasus did too) Pythagoras theorem, to show how some numbers can not be written as fractions (in modern terminology these are called irrational numbers) Imagine a simple square, each side 1cm in length. How long is the square's diagonal? Using Pythagoras theorem (a squared + b squared = c squared) Then the diagonal is 1 squared + 1 squared so the length of the diagonal is the square root of two.

Now, if you have a calculator at hand find the square root of two. The display will say 1.414213562. This number in of itself wasn't a problem for the Pythagoreans. They just had to figure out what two whole numbers made a fraction that produced this number. There was just one little problem no matter how hard they tried the Pythagorean cult could not find two whole numbers whose ratio produced this number. The only (correct) conclusion was….there is NO ratio that will produce √2. the decimal will go on for ever and ever. It is, in other words, an irrational number.

The maths bit

We can use what is called a proof by contradiction to show √2 is irrational. Firstly, let's suppose √2 is not an irrational number (i.e a rational number).  Then two whole numbers as a fraction will produce this number so we can show this as √2  = a/b where a and b are whole numbers, and b is  not zero.

We additionally assume that this a/b is simplified to the lowest terms, in order for a/b to be in its simplest terms, both a and b must be not be even. One or both must be odd. Otherwise, you could simplify. So if √2  = a/b it follows that:

2 = a²/b² (Square both sides) or

  

a² = 2 * b². (Multiply both sides by  b² )

So, the square of a is an even number since it is two times something. From this we deduce that a is an even number. Why? Well, it can't be odd; if a is odd, then a * a would be odd too. (Odd number times odd number is always odd. I can prove this if anyone wishes, it is very simple to do)

If a is an even number, then a is 2 times some other whole number, or a = 2n where n is this other number.

Now, If we substitute a = 2n into the original equation 2 = a²/b², this is what we get:

2	=	(2n)2/b2
2	=	4n2/b2
2*b2	=	4n2
b2	=	2n2.

This means b² is even, (because b squared = 2 times a number squared) from which follows again that b itself is an even number.

And that’s it. We have a contradiction. For why I hear you ask?  Because we started this logical process saying that a/b is simplified to its lowest terms and hence either a or b must be odd. It turns out though that if the √2 is rational then a and b would both be even. So logically √2 cannot be rational.

QED!

Anyhoo. That’s not the main purpose of today. Pythagoras is a dastardly murderer and his ideas are still corrupting the young 2,500 years after his death!