AMC 8 to AMC 10: what actually changes

Most students moving up from the AMC 8 expect the same paper with harder questions. That is part of what actually changes, and not the first part. Before the questions get harder, the paper changes its rules, and one of those rule changes turns the most reliable AMC 8 habit into a way to lose points.

Both papers are 25 multiple-choice questions with five choices each. Almost everything after that is different.

The rules, side by side

• Time. The AMC 8 gives you 40 minutes, which is 96 seconds a question. The AMC 10 gives you 75 minutes, which is 180 seconds a question. Each question gets nearly twice as long.
• Scoring. On the AMC 8 your score is the number you got right, and a wrong answer and a blank are both worth nothing. On the AMC 10 a correct answer is worth 6, a blank is worth 1.5, and a wrong answer is worth 0.
• Where it leads. The AMC 8 does not qualify you for anything after it. The AMC 10 is the first paper on the route to the AIME, which is why its scoring rule matters far more than it looks.
• How many sittings. The AMC 10 runs in two versions, A and B, on different dates, and you may sit both. The AMC 8 is one sitting.

The first two are the ones that change how you should sit the paper, so they get the rest of this page.

The habit that flips: filling every bubble

On the AMC 8 there is no penalty. A blind guess between five choices is worth $\tfrac15$ of a point on average and a blank is worth nothing, so answering everything is simply correct. Every good AMC 8 student learns to fill the last few bubbles in the final minute, and they are right to.

On the AMC 10 the same blind guess is worth $6 \times \tfrac15 = 1.2$ points on average, and a blank is worth 1.5. The guess is now the worse option. Cross out one choice and a guess is worth exactly 1.5, a tie. Cross out two and it is worth 2, and guessing is right again.

Put numbers on the last minute. Ten bubbles filled at random land two on average, for 12 points. The same ten left blank are 15 points, guaranteed. A student who carries the AMC 8 habit up the ladder gives away 3 points on average, half a correct answer, for nothing at all.

The full arithmetic of what blanks are worth against a target score is in how many questions you need for the AIME.

The clock gets longer, and so do the questions

180 seconds instead of 96 sounds like relief. It is not, because the questions grow to fill it. Most AMC 8 questions are one idea: see it, apply it, done. From the middle of the AMC 10 onward, a typical question needs two ideas chained together, and the hard part is noticing the second one.

The clearest way to see the change is to watch what stops working, and the thing that stops working is listing. Three examples.

Counting. How many three-digit numbers have digits that strictly increase, like 147 or 389? The AMC 8 instinct is to list them by first digit. That works, slowly, and the AMC 10 version of the question will have bigger numbers in it. The move that scales is to notice that choosing three different digits from 1 to 9 fixes the number completely, because there is only one way to put three digits in increasing order. So the count is $\binom{9}{3} = 84$, and zero never appears because the only place it could go is first. That is constructive counting: describe how to build each object exactly once, then count the choices at each step.

Remainders. What is the remainder when $2^{100}$ is divided by 7? Nobody computes $2^{100}$. The powers of 2 leave remainders 2, 4, 1, 2, 4, 1 on division by 7, repeating every three steps, and $100 = 3 \times 33 + 1$, so the answer is the first entry of the cycle: 2. An AMC 8 question might ask for the last digit of a large power, which is the same idea with 10 in place of 7. The AMC 10 expects you to run it with any divisor, and that is what modular thinking is.

Structure instead of simulation. In the hundred lockers problem, student $k$ toggles every $k$-th locker, and the question is which lockers end open. You can simulate the first twenty and notice squares. The AMC 10 wants the reason. A locker is toggled once for each of its divisors, divisors come in pairs $d$ and $n/d$, and a pair only collapses into one divisor when $d = n/d$, which happens exactly when $n$ is a perfect square. So the lockers with an odd number of toggles are the squares, and 10 of the hundred end open. The simulation finds the pattern. The reason tells you it still holds at locker 10,000, where no simulation by hand will go.

In all three, the AMC 8 method still reaches the answer. It reaches it in time only when the numbers are small, and the AMC 10 chooses numbers that are not.

What a practice score means now

On the AMC 8, 15 right is a score of 15, and nothing else about the paper matters. On the AMC 10, 15 right is several different scores depending on what happened to the other ten. Fifteen right and ten blank is $6 \times 15 + 1.5 \times 10 = 105$. Fifteen right and ten wrong is 90. The same knowledge, fifteen points apart.

So when you sit your first AMC 10 practice paper, score it in three columns: right, blank, wrong. The wrong column is the one to shrink first. It is the cheapest to fix, because most of it is habit rather than mathematics, and every wrong answer turned into a blank is 1.5 points you did not have before.

You are not meant to finish

Plenty of AMC 8 students reach question 25 with time to spare. The AMC 10 is built so that most of the people sitting it do not solve every question, and the scoring rule pays you for knowing which ones to leave. Reaching the end is no longer the goal. Spending your 75 minutes on the questions you can actually convert is. If the last page keeps arriving with no time left, why you run out of time on the AMC is about exactly that.

What to do in the first month

1. Sit one AMC 10 practice paper under the real clock and score it in three columns. That tells you which change is costing you the most.
2. Retire the last-minute bubbles. Practise with a fixed rule: blank unless two choices are crossed out. It will feel wrong for a week, because the AMC 8 trained the opposite reflex.
3. For every question you solved by listing, ask what you would do if the numbers were ten times bigger. That question is the bridge from the AMC 8 method to the AMC 10 one, and it is how you find the counting or the remainder argument hiding underneath.

None of this needs a new syllabus first. It needs the same mathematics used under different rules. The diagnostic is the fastest way to see where you stand before you start, the daily problem posts a new one every day in three tiers with no account needed, and the technique pages are the vocabulary the second idea in each AMC 10 question is usually drawn from.