If you can't solve a problem, then there is an easier problem you can solve: find it. - George Pólya
This is probably the most important problem solving technique there is. Students, try this if you get stuck! Note that there are many ways to apply this technique, one is to reduce or increase the number of variables or their values. Another is to substitute variables for specific values. A third is to add or remove constraints. Always look for ways that make it easier to show what you want to show. Do not disregard the new problem for being easy until you have solved it. You typically know whether you want to prove or disprove with some experiments and otherwise, you'll have to work in parallel. See Polya's book for more information.
If, therefore, any be unhappy, let him remember that he is unhappy by reason of himself alone. - Epictetus
In mathematics you don't understand things. You just get used to them. - John von Neumann
When introduced at the wrong time or place, good logic may be the worst enemy of good teaching. - George Pólya
Some things are in our control and others not. Things in our control are opinion, pursuit, desire, aversion, and, in a word, whatever are our own actions. Things not in our control are body, property, reputation, command, and, in one word, whatever are not our own actions. - Epictetus
I was gratified to be able to answer promptly. I said I don't know. - Mark Twain
The most difficult subjects can be explained to the most slow-witted man if he has not formed any idea of them already; but the simplest thing cannot be made clear to the most intelligent man if he is firmly persuaded that he knows already, without a shadow of doubt, what is laid before him. - Leo Tolstoy
We're blind to our own mistakes. (so please tell me)
Get your facts first, and then you can distort them as much as you please. - Mark Twain
The most exciting phrase to hear in science, the one that heralds new discoveries, is not 'Eureka!' (I found it!) but 'That's funny ...' - Isaac Asimov
Choose a job you love, and you will never have to work a day in your life. - Confucius
I am not young enough to know everything. - Oscar Wilde
People do not lack strength, they lack will. - Victor Hugo
the difference in mind between man and the higher animals, great as it is, certainly is one of degree and not of kind. - Charles Darwin
Every generation laughs at the old fashions, but follows religiously the new. - Henry Thoreau
Give a man a fire and he's warm for the day. But set fire to him and he's warm for the rest of his life. - Terry Pratchett
and blatantly my own,
Reason for emotions, not with them.
Strive for the values of your future self.
Like laws, our "rights" are but moral heuristics.
Being an idealist does nothing for the world.
I like the winter, it makes us appreciate the summer.