- 24 Mar '13 22:56Readers with mathematical interests may be interested in this thread.

While it's been a long time since I did mathematics, I long have regarded

mathematics as an oasis of (comparative) rational order in a chaotic world.

Mathematics seemed 'fairer' to me than the outside world. Even a child

who (as I was) might be persistently treated as intrinsically inferior on

account of the prejudices of class, nationality, race/ethnicity, or gender

*could* (as I learned to do) use the universal language of mathematics

to speak on equal terms with the (competent) adult world. It seems

empowering for a young person to use mathematics to show an adult

in authority (even with a Ph.D.) that one's mathematical insight is right.

During a recent interval in my domestic duties, my mind unexpectedly

turned toward a mathematical problem that I had solved as a young

student, and I was pleased when, having forgotten what I had done,

I was able to solve it again. While now almost all of my mathematical

knowledge might have 'gone with the wind', I felt happy in knowing that

I had kept a spark of mathematical insight (perhaps an ember next?).

What kind of boy or girl (yes, some girls love to do mathematics) would

do (as I did) 'maths for fun'? In many cases, a child with a parent who

had a strong mathematical background, but that was not true in my case.

(My parents knew little of mathematics.) In my case (as with a few of

my acquaintances), I was an intellectually curious child with hardly any

materially privileged opportunities. Many more privileged children would

play with their chemistry sets or take piano lessons. Lacking a chemistry

set and a piano (or any musical instrument), I knew that mathematics was

cheap. My father liked it because he would have to spend (or waste)

hardly any money on my education, and he hoped that maths would keep

me out of trouble. So after an unusually late start in maths, I found that

maths was a natural language for me, very easy to learn up to a certain

level, and so I enjoyed solving mathematical problems for recreation.

To be continued... - 25 Mar '13 00:08

We all enjoy solving problems.*Originally posted by Duchess64***Readers with mathematical interests may be interested in this thread.**

While it's been a long time since I did mathematics, I long have regarded

mathematics as an oasis of (comparative) rational order in a chaotic world.

Mathematics seemed 'fairer' to me than the outside world. Even a child

who (as I was) might be persistently treated as intrinsicall ...[text shortened]... el, and so I enjoyed solving mathematical problems for recreation.

To be continued...

Some do it in their heads

or with a pen or with chalk

and a blackboard.

Or some with a gun.

What's your point? - 05 Apr '13 00:14 / 2 editsAfter taking a break last week, I am too busy with work now to have much time

to write here. So I shall just present a recreational mathematical problem

that the readers might enjoy attempting to solve. This problem requires little

mathematical knowledge, just a bit of mathematical insight. I don't know of a

published source for this problem. When I was a young student playing around

with number theory, this problem occurred to me, and I thought that I solved in

an elegant way. Later I met a professor of mathematics, who agreed with me

that it was a quite interesting problem; he (as an algebraist) attempted to solve

it with an approach that seemed far removed from classical number theory.

Let Z be the set of (linear combinations) of this form {z: z = Ax + By}

A and B are relatively prime constant integers, A > 1, B > 1

x and y are variables, non-negative integers.

What is the greatest integer k that's not a member of Z?

(Write the outline of a proof if you can.)

For example, if A=9 and B=5, then k would be 31.

31 is the greatest integer that's not a member of {z: z = 9x + 5y}

I would suggest that any professional mathematicians here allow amateurs

the opportunity to solve this problem. - 05 Apr '13 00:18

Help Soothfast.*Originally posted by Duchess64***After taking a break last week, I am too busy with work now to have much time**

to write here. So I shall just present a recreational mathematical problem

that the readers might enjoy attempting to solve. This problem requires little

mathematical knowledge, just a bit of mathematical insight. I solved this

problem again recently.

Let Z be the set o ...[text shortened]... is the greatest integer k that's not a member of Z?

(Write the outline of a proof if you can.) - 05 Apr '13 00:39Here's another recreational mathematical problem. Again, this problem requires

little mathematical knowledge, just a bit of insight. (I solved it in a minute or two.)

If I recall correctly, this problem is a special case of a problem that Ramanujan

mentioned to his friend, G.H. Hardy.

Let's suppose there's a street where the addresses of houses are numbered

from 1 to N: 1, 2, 3 ... N. Let's suppose that you reside at the address M.

(All houses are on the same side of the street.) You notice that the sum of

the addresses on your left equals the sum of the addresses on your right.

That is, 1 + 2 + 3 +...+ (M-1) = (M+1) + (M+2) + (M=3) +...+ N.

If there are at least 50 houses on the street, what's your lowest possible address?

That is, if N>=50, what's the mimimum possible value for M?

You could write a computer programme to solve this problem, but that's cheating!

Just solve it using pencil and paper; you should not need to use a calculator. - 05 Apr '13 00:43 / 1 editAnd if you would like an extremely simple recreational mathematical problem,

you could try this (the answer was almost instantly intuitively obvious to me):

Let X1 + X2 + X3 + ... + Xn = 2016, where each Xi is a nonnegative integer.

What is the maximum product of X1 * X2 * X3 * ... * Xn?

If there seems to be enough interest, I could post some brief solutions to

these mathematical problems sometime next week. - 05 Apr '13 20:24

A theoretical physicist, even these days, may still do most work with paper and pencil, since "number-crunching" is not necessarily the name of their game. Data analysis, on the other hand, will invariably involve a computer.*Originally posted by Eladar***Do you do them? Or do you operate a computer that does them for you?**

If you are doing them, then you are functioning in the stone age. - 05 Apr '13 20:52

Would it be Ok to approach this collaboratively, suggesting ideas?*Originally posted by Duchess64***Here's another recreational mathematical problem. Again, this problem requires**

little mathematical knowledge, just a bit of insight. (I solved it in a minute or two.)

If I recall correctly, this problem is a special case of a problem that Ramanujan

mentioned to his friend, G.H. Hardy.

Let's suppose there's a street where the addresses of houses are ...[text shortened]... cheating!

Just solve it using pencil and paper; you should not need to use a calculator.

If so, one idea I have had is that if the sum of 1 to (M-1) (I'll call it the LHS sum) is equal to the sum of M+1 to N (the RHS sum), then the quantity ((RHS sum) - (M*N)) is equal to summation of 1 to (N-M). It subtracts M from each term on the RHS.

Even if the above is correct and on the right track, I am not sure how to proceed. Subtracting N*M from the LHS brings the two sides into equality: sum (1 to M-1) - (M*N) = sum (1 to N-M). So? - 05 Apr '13 21:48

"Would it be OK to approach this collaboratively, suggesting ideas?"*Originally posted by JS357***Would it be Ok to approach this collaboratively, suggesting ideas?**

If so, one idea I have had is that if the sum of 1 to (M-1) (I'll call it the LHS sum) is equal to the sum of M+1 to N (the RHS sum), then the quantity ((RHS sum) - (M*N)) is equal to summation of 1 to (N-M). It subtracts M from each term on the RHS.

Even if the above is correct and on t ...[text shortened]... M from the LHS brings the two sides into equality: sum (1 to M-1) - (M*N) = sum (1 to N-M). So?

--JS357

When I created this thread, my aim was to encourage people who normally

don't have to do mathematics to have some fun in attempting something unusual

(in stepping outside their 'comfort zone'. If you feel that you could still enjoy

solving a problem collaboratively, I don't mind. Again, I would just ask that you

don't write a computer programme because that's against the spirit of the problem.

Of course, I already know how to solve these problems (which I first did on my

own as a young student). Given that I don't wish to appear too much like a nanny

here, I would prefer not to oversupervise you. I would like to encourage you to

take some steps on your own even at the risk of falling down. You learn maths

by attempting to do it, not just by reading about someone else has done it.

You learn chess by playing, not just by reading books about how to play chess.

Often, when I have explained how to solve a problem to someone who had been

perplexed by it, I have been told, "It's all so simple! Surely what you did must

be trivial. I must have known how to solve it all along; I just didn't express it."

For whatever it's worth, I don't come from a mathematically-inclined family;

my parents had little education in mathematics. Initially, I was stereotyped at

school as extremely poor at mathematics. A teacher said that I obviously was

hopeless, and the children from more educated and privileged families liked to

put me down. That was because I never had been taught anything about even

basic arithmetic. So one day my father finally decided to teach me how to add,

subtract, multiply, and divide--which I learned that day (it took a few hours).

He demanded that I be able to perform basic arithmetic almost instantly and

always correctly. He would drill me by shouting, for example, "What's 13 x 13?",

and I would reply, '169'. As usual, the implicit threat of a beating hung over me.

Most middle-to-upper class Western children, I suppose, would have found the

"throw me into the deep end and sink-or-swim" conditions unbearable. But I

was a natural swimmer. When school resumed, everyone else seemed shocked

that I could perform basic arithmetic more quickly and accurately than every

other student and my 'teacher'. Of course, then there came strong pressure from

the school authorities for me to conceal my intelligence as much as possible.

Someone from my background was expected to stay near the bottom, not to

surpass the children from much more privileged and supportive families.

Those children had more opportunities and diversions (e.g. piano lessons), so

they never had to seek fulfillment in the rather bleak circumstances around me

and they presumably never shared my delight in drawing much from little.

Personally, I could not understand why everyone else did not find 'school'

mathematics as effortless as I did. After all, the basic facts of arithmetic (e.g.

the multiplication tables) are well-known and invariant, so why should anyone

ever make any errors? So I did not consider myself as anyone special at maths.

Later, I was brought to the attention of some mathematicians, who told me that

I obviously was extraordinary and who attempted to convince my parents of it.

Of course, my parents found it extremely hard to believe that I could ever be

better than ordinary, if that, in doing anything. In contrast to Ruth Lawrence,

for instance, I was a rather indolent and unambitious student. A few years after

I learned arithmetic, I learned number theory (at an undergraduate level), but

I had no one around me with whom to communicate my mathematical thoughts.

So my isolated circumstances hindered my development. Ramanujan thrived

in isolation, but, of course, I could hardly approach his level of natural genius. - 06 Apr '13 01:05 / 1 edit

Based on what you say about your father's motivational techniques, I can understand your aversion to brute force solutions to problems.*Originally posted by Duchess64***"Would it be OK to approach this collaboratively, suggesting ideas?"**

--JS357

When I created this thread, my aim was to encourage people who normally

don't have to do mathematics to have some fun in attempting something unusual

(in stepping outside their 'comfort zone'. If you feel that you could still enjoy

solving a problem collaboratively, I d , but, of course, I could hardly approach his level of natural genius.

If this were my poser, I would announce the correct answer up front (assuming there is but one) and ask for both the most elegant brute force iterative process of identifying it and the most elegant mathematical proof.