When Should You Guess on the AMC 10/12? A Mathematical Guide to Scoring Strategy

Summary

Knowing when to guess on the AMC 10/12 can help you earn a higher score. If you can eliminate two or more answer choices, guessing is usually a good idea. If you have no idea which answer is correct and all five choices seem equally likely, leaving the question blank is better. When four choices remain, guessing and leaving the question blank are equally good in terms of points on average. The key is to use what you know, eliminate choices whenever possible, and avoid spending too much time on a single difficult problem.

Leaving the question unanswered earns 1.5 points. Therefore, guessing is better when

or equivalently,

In other words, guessing is worthwhile when your chance of being correct is greater than 25%.

If n answer choices remain and each is equally likely to be correct, your chance of guessing correctly is 1/n. Your average score from guessing is therefore

The table below shows how this works in practice.

Note: Guessing gives you a chance to earn 6 points, while leaving the question blank guarantees 1.5 points.

These calculations assume that all remaining choices are equally likely to be correct. In practice, your reasoning and how confidently you eliminate choices also matter.

3. What Is an Educated Guess?

An educated guess is not simply picking an answer at random. It means using what you know to eliminate incorrect choices, even if you cannot solve the entire problem.

On the AMC 10 and AMC 12, you can often eliminate answer choices by using the following techniques:

• Check signs, parity, and size. Determine whether an answer must be positive or negative, even or odd, or within a certain range.

• Try the answer choices. For some algebra problems, substituting the choices directly is faster than solving the equation from scratch.

• Use divisibility and modular arithmetic. Number-theoretic properties can rule out choices that do not satisfy necessary conditions.

• Estimate the answer. Approximation and simple upper or lower bounds can help eliminate unreasonable choices.

• Look for necessary conditions. Even if you cannot finish a proof, you may be able to identify properties that the correct answer must satisfy.

These techniques help you make better guesses and become a more efficient problem solver.

Keep in mind that eliminating an answer choice only helps if you have a good reason to believe it is incorrect. A guess based on shaky reasoning may not be as reliable as you think.

4. When Should You Guess on the AMC 10/12?

Here is a practical guide for contest day.

• Five choices remaining: If you have no useful information and all choices seem equally likely, leave the question blank. Random guessing gives you a lower average score.

• Four choices remaining: If all four choices seem equally likely, guessing and leaving the question blank give you the same points on average. If you have a good reason to favor one choice, guessing may be better.

• Three choices remaining: If you have reliably eliminated two choices, guessing is worthwhile.

• Two choices remaining: If you have narrowed the answer down to two plausible choices, guessing is generally a good idea.

• One choice remaining: If your elimination is logically sound, select the remaining answer.

The main point is that you should decide whether to guess based on how likely you are to be correct, not just how many choices you have eliminated.

For example, if you eliminate two choices and the remaining three are equally likely to be correct, guessing gives you an average of 2 points, compared with 1.5 points for leaving the question blank. But if your reasoning is unreliable, your actual chance of being correct may be lower.

5. Time Management: The Most Important Strategy

Guessing is only one part of a good contest strategy. You have 75 minutes to solve 25 problems of varying difficulty. Spending too much time on one difficult problem can prevent you from earning points on other problems you could solve.

A practical approach is to:

1. Solve the problems you know first. Secure the points available on questions you can solve confidently.

2. Move on when a problem takes too long. Mark it for review instead of letting it consume too much time.

3. Return to difficult problems when time permits. Try a different approach and look for opportunities to eliminate answer choices.

4. Make educated guesses when appropriate. Use mathematical reasoning to decide whether guessing makes sense.

5. Save time for checking. Review your calculations, verify your answers, and make sure you have not overlooked any solvable problems.

During practice, aim to solve as many problems correctly as possible within the 75-minute time limit. Ideally, work toward solving all 25 problems accurately. This will help you build the mathematical skills and speed needed for stronger contest performance.

On contest day, however, your goal is to earn the highest score possible by combining accuracy, smart problem selection, effective time management, and educated guessing.

6. Final Takeaway

The AMC 10 and AMC 12 reward strong mathematical skills, but a good guessing strategy can also make a difference.

Remember these simple rules:

• Three or fewer choices remaining: Guess, assuming the remaining choices are equally likely to be correct.

• Four choices remaining: Guessing and leaving the question blank give you the same points on average if all four choices are equally likely.

• All five choices remaining: Leave the question blank if you have no useful information and must guess randomly.

Eliminating answer choices is helpful, but only when your reasoning is reliable. Do not spend too much time trying to eliminate choices on a difficult problem.

Ultimately, the best strategy is to build strong mathematical skills, solve the problems you know how to handle, manage your time wisely, and use educated guesses when they give you a reasonable chance to earn more points.