How can one avoid silly mistakes? [closed]

I find that it is oftentimes that I am hacking away, writing an answer only to realize that a silly mistake I made quite some lines above has done me in and I have to go back and redo a significant chunk. This wastes quite a bit of time.

Even worse is when I have hacked away at a problem and submitted it, only to realize some time later that there’s a fatal error in the body… and a whole mess ensues. This wastes my time and the time of people who end up reading it before I realize.

In fact, how can one avoid silly mistakes in general?

In time-based competitive examinations, candidates do not always have enough time to mull over what they have just done and fix mistakes. Not only that, a silly mistake in such a high stress, often high stakes situation is even more likely. And for those prone to making such errors, the stress of looking out particularly for silly mistakes while solving wastes a lot of time.

Silly mistakes range from a missed sign ($$\pm$$), to a botched expansion, to mixing up coefficients, to falling to abuse of notation, to simply having misread the question… simply put, any petty but fatal error that you commit despite clearly knowing better. (Or, maybe understanding better is a more accurate expression).

Looking forward to responses.

• You can't. Such is the nature of mathematics. All you can do is repeat a mistake enough so you recognize it faster. – Asaf Karagila Sep 29 '19 at 18:22
• Occasionally one can recognize an (otherwise subtle) error in reasoning from the incorrect conclusions drawn, sometimes called "the argument that proves too much." In broader settings one should check calculations with ball park estimates and other means (e.g. differentiating an indefinite integral to ensure the algebra is right). – hardmath Sep 29 '19 at 18:55
• There is no royal road. – Jack Sep 29 '19 at 20:56
• What I learned from my own math competition experiences is to check your work as much as you can before submitting it. Rework it manually from scratch without looking at your work from the first time, and if there is a divergence, determine why. If you can verify your answer in a completely different way, that also tends to help. Of course these aren't foolproof, and it helps if you take some time between the reworking and the original working so as to not somehow bias yourself. Beyond that, it's mostly practice. You're never too old or too experienced to learn - mistakes are a part of that. – Eevee Trainer Sep 29 '19 at 22:44
• I suggest downloading Latex to your home computer. When you feel you have an answer, typeset it at home, render it there, see if you still believe it. That's what I did. – Will Jagy Sep 30 '19 at 15:25