Each problem that I solved became a rule, which served afterwards to solve other problems.

It's not that I'm so smart, it's just that I stay with problems longer.

It is the duty of all teachers, and of teachers of mathematics in particular, to expose their students to problems much more than to facts.

The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would be solver.

One of the big misapprehensions about mathematics that we perpetrate in our classrooms is that the teacher always seems to know the answer to any problem that is discussed. This gives students the idea that there is a book somewhere with all the right answers to all of the interesting questions, and that teachers know those answers. And if one could get hold of the book, one would have everything settled. That's so unlike the true nature of mathematics.

One of the big misapprehensions about mathematics that we perpetrate in our classrooms is that the teacher always seems to know the answer to any problem that is discussed.

If there is a problem you can't solve, then there is an easier problem you can't solve: find it.

It isn't that they can't see the solution. It is that they can't see the problem.

A GREAT discovery solves a great problem but there is a grain of discovery in any problem.

The measure of our intellectual capacity is the capacity to feel less and less satisfied with our answers to better and better problems.

Theorems are fun especially when you are the prover, but then the pleasure fades. What keeps us going are the unsolved problems.

When I am working on a problem, I never think about beauty but when I have finished, if the solution is not beautiful, I know it is wrong.

The tantalizing and compelling pursuit of mathematical problems offers mental absorption, peace of mind amid endless challenges, repose in activity, battle without conflict, refuge from the goading urgency of contingent happenings, and the sort of beauty changeless mountains present to senses tried by the present day kaleidoscope of events.

The theory of numbers is particularly liable to the accusation that some of its problems are the wrong sort of questions to ask. I do not myself think the danger is serious; either a reasonable amount of concentration leads to new ideas or methods of obvious interest, or else one just leaves the problem alone. "Perfect numbers" certainly never did any good, but then they never did any particular harm.

There are problems to whose solution I would attach an infinitely greater importance than to those of mathematics, for example touching ethics, or our relation to God, or concerning our destiny and our future; but their solution lies wholly beyond us and completely outside the province of science.

He'd met other prodigies in mathematical competitions. In fact he'd been thoroughly trounced by competitors who probably spent literally all day practising maths problems and who'd never read a science-fiction book and who would burn out completely before puberty and never amount to anything in their future lives because they'd just practised known techniques instead of learning to think creatively.

Christians remind me of schoolboys who want to look up the answers to their math problems in the back of the book rather than work them through.

We've lost something that's been with us for so long, and something that drew a lot of us into mathematics. But perhaps that's always the way with math problems, and we just have to find new ones to capture our attention.

I hate the idea that, when it comes to books and learning, hard is often seen as the opposite of fun. It's strange to me that we should be so quick to give up on a book or a math problem when we are so willing to grapple, for centuries if necessary, with a single level of Angry Birds.

All human states are organic brain states - happiness, sadness, fear, lust, dreaming, doing math problems and writing novels - and our brains are not static.

In the end, climate change is a math problem.

What's clarity like? Try to remember that funny feeling inside your head when you had math problems too difficult to solve: the faint buzzing noise in your ears, a heaviness on both sides of your skull, and the sensation that your brain is twitching inside your cranium like a fish on the beach. This is the opposite sensation of clarity. Yet for many people of my era, as they aged, this sensation became the dominant sensation of their lives. It was as though day-to-day twentieth century living had become an unsolvable algebraic equation.

We sat looking out at the ocean. There was just so much of it, and it never failed to take my breath away. Looking at the ocean gave me the same sensation I'd get staring at a sky full of stars- that I was small. Like the way a math problem reveals its undeniable truth, I knew when I stared into this sort of endlessness that my life didn't count for much of anything. And knowing that, that I was nothing but a speck, I felt pretty lucky for all that I had.

The problem with cinema nowadays is that it's a math problem. People can read a film mathematically; they know when this comes or that comes; in about 30 minutes, it's going to be over and have an ending. So film has become a mathematical solution. And that is boring, because art is not mathematical.

The Tour (de France) is essentially a math problem, a 2,000-mile race over three weeks that's sometimes won by a margin of a minute or less. How do you propel yourself through space on a bicycle, sometimes steeply uphill, at a speed sustainable for three weeks? Every second counts.

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