Rambling puzzles…Number plates, shoelaces and you.

Square root of x formula. Symbol of mathematics.

Image via Wikipedia

I recently read 2 rather interesting questions online. The first was this:

Mr Smith has 8 children, and each one is a different age. His entire family is in the car, when his eldest, aged 9, shouts out “Daddy! That number plate has only two numbers, and each number is repeated twice! And the number is divisible by each of our ages!” “You’re Correct!” Mr Smith shouts as he whips out his iPhone to check. “and look, the last two digits are my age!”

What is Mr Smith’s age?Actually the question wasn’t like that, it gave you choices and you had  to say which of those numbers were not the age of one of his kids, but this question is a bit harder and more interesting.
The second question:there are 30 shoelaces in a closed box with all of their ends sticking out (so 60 ends). You tie one end to another, and keep doing this until each end is connected to one other.
How many ways are there of doing this?
What is the expected numberof loops?This was on the “Mind Your Decisions” Blog, and I gave a comment which got beautifully shot down by a fellow commentator, And I’m still uncertain who was right.
But anywho, have a go at the puzzles and see if you can solve them.
The first question I don’t know if it’s possible to answer without extra info. If it isn’t, try work out what extra info you need.
As for the second, the only decent suggestion I have is you start with a smaller number and see what happens.
I also recently discovered that Cambridge University has a very different teaching approach to mathematics. They arrange for 2 students to meet up with a lecturer to go over mathematics that they’ve been doing. I’m quite jealous of this! And yet I wonder if I’d actually take the opportunity…doubtless there are people somewhere who will hear about Auckland University and be amazed I haven’t taken more opportunities that have been offered to me, yet from my point of view, they don’t seem like options.
One thing I’ve always been told to do is go and speak to my lecturers. (In fact, I’ve told people to do it myself). Yet I never have. I always think “what would I talk to them about?” I have one query for one of my lecturers, but it seems incredibly trivial and it has no point other than to give me something to talk about with him for about 43 seconds (it takes about 13 seconds to say hello, how are you etc normally). And then? Do I just leave or what? This is why I would like the chance to have a supervisor, but the truth is I do have the chance, I’m just not taking it.I was aiming to speak to one of my lecturers this last week, but have been unable to attend uni this last week and a half…so I think I’m going to aim to go and speak to some of them next week. With some good questions. Hopefully.
I’ve been reading a fair amount of mathematics recently,online and print, about mathematics and mathematicians. There are a variety of mathematicians, some are ‘gifted’ (or ‘genius’) and others persevere and work hard at it – and of course there are varying degrees of both in some mathematicians, and some great mathematicians have neither. But no matter what ‘category’the mathematician falls into, one key characteristic is passion, passion for mathematics. Another, a willingness to forge a new path that no ones been down before.
It is this second that strikes me most. Too often we do what we are expected to do, and not what we want to do. The joy is doing what we want. If we feel forced into doing it, there is not much joy there.
Some mathematicians will tell you to memorise important theorems and proofs – that is the way to be a good mathematician. Others will tell you to discover the proofs and theorems yourself – that is the way to be a good mathematician. And yet others will tell you find a good mathematician and be their ‘apprentice’ – that is the way to be a good mathematician.
But I guess it all comes down to us as individuals. What do you want to do? How do you want to do it? Forget about the marks or opinions,what do you want to do?
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1089 (, 495 and 6174) and all that

Take any 3 digit number where the first and last digit areat least 2 apart. (1st number)
Reverse it. (2nd number)
Minus the smaller number from the larger one. (3rdnumber)
Reverse it (4th number)
Add the 3rd and 4th number together.
Your result is 1089.
 This problem i first discovered in the book “1089 and all that”…an awesome little book!
Now, take any 3 digit number, with the only requirementbeing that all 3 numbers are not equal (so the number 001 is allowed)
Now, using these three digits, rearrange the numbers to makethe smallest and biggest possible numbers.
Now subtract the smaller number from the larger one.
Repeat this process. After a few steps you will reach 495.
Now do the same process for 4 digit numbers….you get 6174.
The great thing about these processes is they seem magicalto someone who has never heard of it before. Try the 1089 trick on a 10 yearold, you’ll convince them you’re magical!
These ‘tricks’ are – to me – some of the most interesting ‘tricks’around – well, when it comes to mathematics at least.  So what I thought I’d do in this post isexplore these numbers and why these methods work, and then see what othernumbers we can find etc.
So starting off with 1089:
Lets say your three numbers are x,y,z in that order. Lets alsosay that x is greater than z. so our two numbers are xyz and zyx. Subtracting thesegives:
(x-z) (y-y) (z-x)
Now, as z is less than x, we need to ‘borrow’ a 1 from they, so we actually have
(x-z)(y-1-y)(10+z-x)
But we also need to ‘borrow’ a 1 from the x
(x-1-z)(10+y-1-y)(10+z-x)
As we’re working mod 10, and if we let x-z =c, then we have
( c-1 ) (9) (10-(x-z))
Which is
( c -1) (9) (10-c)
Now we reverse this number
(10-c) (9) ( c-1)
+             ( c-1 )(9) (10-c)
=             ( 9)   (18) (9)
=             (9+1) (8) (9)
=             1089
This is always true.
What about 495?
Lets take the number x,y,z where x>y>z
Then we have
x    y      z
         z     y      x
= (x-z)(y-y)(z-x)
Again, z<x so need to ‘borrow’ a 1 from the y, whichmeans we’ll need to borrow a 1 from the x. again, let x-z=c
(c-1)(9)(10-c)
If c were 1, we would have the number 990 and 099 for thenext round. If c were 0, it would mean that x=z and as x<y1
If c >5, then 10-c<c-1, so we will first look at when1<c<6
the biggest number will be
(9) (10-c)  (c-1)
And the smallest
(c-1) (10-c) (9)
Now, subtracting them gives
(9-c+1) (10-c-10+c) (c-1-9)
Again, we have to ‘borrow’ so
(9-c) (9) ( c )
With c being 2,3,4 or 5
If we did the same thing when c>5 , we would end up with( c ) , (9 ) , (11-c)
 With c being, 6,7,8or 9.
If c is 2
We have 792, so
972-279=693, which is what we get if c is 3.
693 gives us
963-369=594, which is what get if c is 4…and  if c is 5, we get the number 495, which isthe magic number!
To confirm, 594 is 954-459=495.
If c is 6
We have 695 which is
965-569
396 which is the same as 693
If c is 7
We have 794 which is
974-479=495, which is the magic number
If c is 8
We have 893 which is
983-389=594 the magic number almost
If c is 9
We have 992 which is
992-299=693 which we’ve already done.
Quickly looking at c=1
990-099=891 which is
981-189=792 which we’ve already done.
This shows that it works for all numbers, unless all 3digits are the same.
A similar method can be used to show that for four digits,you get the number 6174, and this number is called the Kaprekar number.
What interests me is: what happens if you have n digits? Whathappens if you use repeated different operations (so using addition andsubtraction, but in different orders…maybe you subtract, subtract, add,subtract, subtract, add…)?
You could classify a sequence of operations as a binary operation.You could do operations on reversed numbers, or operations on biggest numberthat can be made from those digits and the smallest.
Any of these methods could yield some interesting observationsin number theory. Applications of these observations could be used in chaostheory or cryptography….but the point of doing number theory is that there areseldom applications!
In a way, these tricks remind me of collatz conjecture. EvenWikipedia thinks so, as the wiki page on 495 and 6174 both link through tocollatz. Interesting correlations.
I’ll end this post by looking at the Kaprekar routine fornumbers with 2 digits.
xy, with x>y
we have xy-yx=(x-y-1) (10-(x-y))
let x-y=c
we have (c-1) (c)
so we have (c) (c-1)-(c-1)(c)=(c-c+1-1)(c-1-c+10)=9
repeatedly subtracting the smallest possible number from thebiggest possible number from a set of digits for n digits:
digits     result
2              9
3              495
4              6174
5              ?

Do you want a mathematician or a computer?

Recently I have been applying to a couple of jobs for next year, and have beenquite fascinated by the anxiety other people express over the whole applicationprocess. It seems everyone is so intent on being perfect for the job they’reapplying for, that they don’t seem to notice that if they have to ‘change’ whothey are to get the job, the job probably won’t suit them. So they doctor uptheir C.V’s and buy different clothes and attend the interviews readyingthemselves to answer in such a way that their weaknesses become strengths….

The best one I’ve heard from this is: “what is your biggest weakness?”
“I’m dedicated to my work, so I often work straight through a lunch break”.

Yeah right.
Anyways, a Game theorist wrote a lot more on this here : http://mindyourdecisions.com/blog/2008/08/28/job-interviews-you-don%E2%80%99t-have-to-be-perfect/?dhiti=1

an awesome blog to read, and the articles make you think.

My favourite part of interviews are the questions that makeyou think. Asking what I would do in a given situation is boring, but asking mehow to solve an abstract idea – that’s fun.
My girlfriend had an interview a few months ago, and one ofthe questions was:

using square tiles that have an area of 1 unit squared, how can you lay thetiles such that the number of tiles used is twice the number of tiles in theperimeter?

the way people approach this is fascinating. Most of us will start writing outequations, so perimeter of a rectangle is 2 x (width+length) and the area iswidth x length, so, using w,l to denote width, length respectively,

4(w+l)=wl
4+4l/w=l
which isn’t easily solved. It would probably be best to start throwing integersinto the equation, and see what comes out.
So if we define w as 8, we get 4+l/2=l4=l/2 8=l so a square of 8 by 8 works! Right?
Wrong.
We forgot to notice that thecorner tiles are counted twice. The perimeter calculation should actually be2*(w+l-1) (but the area still stays the same).
so
4(w+l-1)=wl
4+4(l-1)/w =l

again, try values of w

W=8
4+(l-1)/2=l
8+l-1=2l
7=l
Which gives a perimeter of 28tiles and an area of 56, which works.
But to do this in an interviewwhere people are stressed out….that’s a bit more tricky. You’ll forget todouble the perimeter. You’ll forget the repeated tiles. And you won’t noticethat if you tile around the equator, you simply need to place 4 rows of tiles,one above the other, to satisfy the requirements…
There are quite a few questionslike this.  
Here’s another one from the blog Ilisted before:

http://mindyourdecisions.com/blog/2011/09/05/monday-puzzle-paying-an-employee-in-gold/

the solution is awesome.

An alternative solution is tocreate a gold cutting device that has 6 knives. Then you just need one cut.
Now, from my limited experience ofinterviews and applications, it seems that most companies claiming to belooking for mathematicians are actually looking for a computer or a softwaredeveloper (which is good news to software developers!). Almost all graduatejobs base part of the application on marks attained at university.  Interestingly, those marks are made up ofexams, tests and assignments. Assignments seldom give more than 20%  towards the final mark. Roughly 40% of theassignments are based on an ability to think like a mathematician.  The rest is rote learning/memorisation or useof a method.  Unfortunately, the examsand tests are also based on this.  This meansthat most courses have at most 10% of the course based on the ability to think,and 90% on being able to memorise proofs and theorems and methods. 
This is all well and good…except for a little thing called the internet. Throughthe use of google, wolframalpha and Wikipedia, that 90% of the course can bedone by someone with no previous knowledge of the course.  I would also guess that they can do it moreaccurately, and faster.
So to base your applicants ontheir university scores seems to me to be a waste of time. If that is all youwanted, you’d be better off getting a software developer or simply using thosethree sites – this way will be much cheaper! If you ever come across a problemyou can’t solve using those sites…well, then there isn’t a high chance yourgraduate with straight A’s could have solved it either…

But enough of that. It’s fairly obvious I have crap marks and think I’m abetter mathematician than those marks show, but that’s my opinion. I likesolving problems, and this is a taster of the next post:

what do the numbers 1089 and 495 have in common?